To feel this mum should take her turn?

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).

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?

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.

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.

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.

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)

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.

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.

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.

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 😂😂

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!

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?

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