To feel this mum should take her turn?

In which case it would have to be arranged in advance.

So just arrange for them to walk from school to the park and then go later.

Then is no need for the other mum to do anything

Why dont you let them both walk from school to the park by themselves and you just pick her up half an hour later than normal from the park?

As a parent who lives near the school and whose child walks home… I would not take time out of my busy day to do a school run so you can leave later. If it is pre-arranged then your child can leave with mine and go to the park. If they don’t want to on the day your child can come home with mine.
but doing a school run I don’t need to do anymore and then stand in the park to supervise your child - wouldn’t even cross my mind!

I love your passive aggressive little digs 😉but you contradict yourself anyway

and as you wrote yourself are sent away if it's not convenient.
That's exactly the point, kids come and go, and sometimes you have other plans.

So of course it would be beyond rude and unreasonable for a mum to decide that her kid would come to your house on a regular basis as a fixed arrangement.

I am not sure why it seems so important for you to feel you are so chill and carefree unlike some people live in more rigid communities
but well done you 😂😂

My children's friends just come over. Either with my children or they turn up and are sent away if it's not convenient. I don't specifically "offer" anything. I made it clear to my children that their friends are normally welcome unless we happen to have something else on which they would know in advance. This isn't a small child play date where you need full on parental supervision. It's a common age for friends popping by to be an informal arrangement. Friends who in not that many months will be in secondary school and just making their own plans and arrangements.

I can't imagine why most parents would object to their house being an occasional refuge for the " we planned to go to the park but now it's chucking it down so can Sophie come to our house for half an hour while her mum gets here on the bus" scenario. If I thought about it I would proactively offer up my house as said refuge, but chances are I wouldn't think about it. I certainly wouldn't be offended if someone asked if this could be done, and I'd probably wonder why they felt they needed to ask.

But I guess some people live in more rigid communities than I do, where everything has to be organised in advance and your child can't go round to someone else's house unless it's pre-arranged via the other parent offering first.

This isn't the mums problem. They don't need supervising in the park and it's your daughter who needs a parent to supervise her route home. Arrange that they can play 1-2 nights a week after school in advance, ask her daughter to keep in touch with you and meet them later so you aren't hanging around.

No you can't "agree" that your child is going to somebody else's house 2 days a week when what you mean is telling the other parent they'll have to have your child😂

You only agree if they OFFERED which no one has and why on earth should they.

Is this a case of yet another parent not being able to say ‘no’ to their child?
The friend is within walking distance to the school and allowed to walk home by themselves. Fair enough.
Your child needs to get the bus and therefore you go and collect her. Fair enough.
When your child or the friend says they’d like to go to the park, you say, ‘Not today love’. Simple 🤷🏻‍♀️
And you have to stop off from the park to buy drinks and snacks? No, no you don’t. ‘Sorry love, not today. Let’s go home and get a drink and a snack.
My son and his friend used to have a habit of asking for a play date straight after school and whilst it’s awkward, me and the other mum made it perfectly clear that play dates/park, had to be arranged previously. Any impromptu request was met with a ‘no’.
As a previous poster said, you’re expecting the other mum to come out and babysit your child. She’s happy for her child to be more independent (understandably for what it’s worth in my opinion)

Yeah this is your problem, sorry. By year 6, kids don’t need supervising at the park. Just tell your daughter e.g Wednesdays and Fridays she can go to the park and you’ll meet her there or at the bus stop, 30 minutes later. Make sure she has a phone so she can call you if she needs and let her know if the park gets cancelled, to wait by the bus stop/other public area for you. Time to cut the strings a bit or you’re both in for a shock next year.

I think i can see what's happening here.

To leave the school they need a permission slip signed. As you have moved you haven't signed a slip because your daughter lives too far away to walk. So you catch a bus to the school then sometimes hang around so that she can go to the park with her friend.

I had a similar situation last year. We live further away so my daughter doesn't walk home either. Her friends were leaving school and going to the park.

How I got around this was signing the slip and allowing her to go to the park on a Friday if the weather wasn't predicting rain. If it did change during the day I'd show up at usual pick up time. If it remained dry I arrive to the park about an hour later than the usual pick up time. That way she gets to go with her friends after school but short enough so she doesn't need to go to anyone's house. As I don't think that's fair on other parents without prior arrangement.

You could try something like this so she has some independence. Or arrange it for a time over the weekend.

But what you can't do is expect the other mother to be there when her daughter lives closer and is allowed to walk home. She has an arrangement with her daughter, you need an arrangement with your daughter.

I know it can be tough but living further away but it's something you need to manage yourself. I'm in the same situation but we've made it work. My daughter gets her independence without putting on anyone else. So everyone is happy.

I mean the usual sense of the word "agree" where you talk to the other person involved (in this case the other parent) and come up with a mutual arrangement. In this case the parent is quite happy for the two girls to play in the park unsupervised so it's not unreasonale to ask if she would be happy for them to play in the house unsupervised (i.e. nothing for the adult to do). if the other parent, for some reason doesn't want the child at her house at all, then obviously all bets are off.

It's pretty common at this age for the children to be "playing out" and in and out of each others houses (if OP's DD lived closer presumably that might be the case for her).

There are absolutely loads of parents post Covid working at home with this age of child left to amuse themselves with strict instructions not to disturb the parent unless an emergency. It's not like a play date with a smaller child whereyou need hands on parental supervision.

Why would the other mum take a turn? She doesn't need her child taking to the park, she can go alone. It's only you who needs someone to accompany your child.
She would only be going to watch your child... Why would she do that?

The OP didn't mention this option: I did. And in the scenario I was presenting, I was assuming that the mums would be talking with each other (as the OP had suggested she wanted to negotiate alternate park supervisions).

At their age it isn't usual to have supervision at the park.

Surely you must have made arrangements in advance if you used to take it in turns with the other mum? Is that not possible now?

You can give your DD permission to leave the school without you at the gate and go to play at the park. You can get your DD a phone. Yes she'll be a bus ride away if she gets hurt, but that is going to be the case the moment you actually give her a smidge of independence. She will still be okay. It's in the nature of not being joined at the hip to an adult. If it's actually serious other people, like her friend, can call you or, in the worst case, get an ambulance. Expecting another mum to come out and supervise your DD because you don't want to and don't trust her or want to put in place the things she needs to be independent is always going to sound unreasonable.

You might also consider - is she really too young to get the bus home if you coach her? 10 year olds are normally pretty capable.

If you're still uncomfortable with your DD doing this, I wonder - Did you chose this school across the city, or were you just unfortunate in school placement? I'm asking because if you chose it I'd suggest focusing on the fact you're getting all the things you chose it for and see this as just a small price to pay for getting that.

I think the OP would have mentioned if she’d agreed with the mother that her daughter could go back to her house every say…

This! Exactly 💯

You're assuming that the OP hasn't cleared this with the friend's mother, who just might be fine with the idea. She also said that her daughter's friend has a key for the house. It might even be that the mother isn't at home. I've no idea. But the OP was asking for suggestions of how to deal with this situation. One is to see if the other mum just might be happy with this happening on a couple of days in the week. Those of you becoming apoplectic about a 9-year girl spending a half hour in their home after school are the ones being ridiculous.

What do you mean "just agree" that your child will go to another person's house twice a week?

I've read this a few times because I feel like I must be misunderstanding your suggestion.
You can't just send your primary aged child to someone else's house every day after school and think that the parent won't be inconvenienced 😳

Let your DD go to the park with her friend. You arrange to collect them later, say 30 mins after school finishes. They don’t need to be watched if they have phones and are near friends house.

Not inconvenienced?! By a primary age kid in my house making noise, mess, using my facilities and in fact just being there? I welcome my children’s friends all the time, but not daily, unreciprocated and on the whim of the children.

Like hell would I be agreeing to an eleven year old kid coming round every evening so her mother wasn’t inconvenienced by her own choices.

I’m always doing favours for my kid’s friends’ parents and uncomplainingly do far more lifts and hosting than any other parent. Even I would not be offering to go to the park to watch your child because you won’t give her a phone, won’t let her leave school alone, you won’t let her be unsupervised and you live far away. Those are your choices and it’s for you to deal with the consequences. I’d have taken turns when it was mutually beneficial ie when both kids needed supervising but not just as a never ending favour to you. I’m afraid these sorts of arrangements change over time, especially when kids start year six and seven.

I thought that was self-evident, OP.

Sorry you found yourself in such a difficult situation with housing. I think a ‘dumb’ phone for your daughter if you can manage it might be a good idea for situations like this. Plus speaking to the school about pre-arranged permission for her to leave without you. Then pre-plan the park outings or other get together. I know it’s nice to be spontaneous but it’s not very practical in your situation (unless you want to be cold in the park).

You're making this so much more difficult than it needs to be.

DD: Mum, we're going to the park with friend now.
YOU: No, we are going straight home today.
DD: URGHHH but we already arranged it!
YOU: Not with me you didn't.
DD: I hate you.
YOU: Oh well. Look, there's the bus, let's go.

As a parent, you need to use boundaries at times. It's not ok for your DD to frequently make plans like this without consulting you first. Speak with your DD and tell her that from now on, her & her friend can only go to the park after school once per week (or frequency of your choosing). The other 4 days are straight home.

This has got nothing to do with the other mum, and everything to do with your lack of parenting/boundary setting.