To feel this mum should take her turn?

Although I can understand why this is frustrating for you, it really isn’t the other mother’s issue. Her child doesn’t need supervision at this time. If yours does then it’s on you to provide it. Or to come up with a solution that suits you. This will soon become less of an issue for several months as it will be dark and cold!
I do understand, my Dd has needed much more supervision over the last few years than her peers ( she’s a teenager) due to medical issues. But that’s our problem not her friends parents. If she goes out an about it’s our responsibility to be the ones that are around on case she needs us. It’s meant hours trailing round shopping centres on my own by that’s just the way it is!

But why would she because she is happy for her daughter to go to the park unsupervised. It isn’t her problem that you want your daughter supervised.

Tell DD that she can only play in the park if she arranges it in advance, ie by phone before school at the latest. That's reasonable since arranging it on the day is inconvenient for you. And consider whether Dd could safely get the bus on her own.

But how did you arrange which parent would take them to the park in the past when you took it in turns? And how would it help if the other mum came to the park for her 'turn' when you have already arrived at the gate anyway?

My daughter is in year 6. She walks home and there is a park on the way home. She can go there to play on the way home if she likes and she lets me know.

if another child wants to go and their mum ends up going with both children that’s between them and nothing to do with me and my dd. The other mum doesn’t have to take turns because you’re not happy to let your dd go unsupervised and she is.

give school a permission slip and give dd a brick phone if you like so she can go by herself with friend and then you collect from the park later. It’s very much a you problem

The issue here is that you feel your daughter needs a certain level of supervision and the other mum doesn't. As the one who 'requires' the supervision then you should do it. The other mum isn't going to do supervision she doesn't see as needed.

The housing situation is really bad luck, but it won't be long until your daughter can get the bus on her own and things will be easier for you.

You’re not answering the question of how it would help if the mum supervised them, since you’d already arrived at the school

Just go later if the park isnt far from the school. She must think her daughter is responsible enough to go herself and is giving her a bit of independence. However if this used to be shared she could drop you a text to see if you mind - i would.

Yes that would make sense except you’ve said several times that these trips to the park are not planned and you only find out about it once you’re already at the school gate so how could this work?

But her daughter doesn’t need watching

Either say no, or let her go with her friend and catch a later bus to collect her?

Sorry OP you are being unreasonable

I agree that it would be nice of her. But not something you can expect.

It's time to prearrange the park visits on regular days. If the weather's bad you can cancel by text and arrive at the school.

I think you actually need to ask the other girl's mum for a favour. That in an emergency your daughter could go to hers or have her daughter call her so there is a nearby adult while you get over. Or if it suddenly pours. On the arranged park days.

Yabu. Was the other mum driving your daughter home half the time before? Its not their fault they live walking distance and you don't. The mum has decided her child can be unsupervised I think you either tell your child no or arrange to collect at a later time

She doesn’t need to take her turn because she doesn’t require a parent to be there. Your daughter could go to the park and you collect her from there or you could ask the mum if your daughter can walk to her house and you collect her from there.

your daughter hasn’t got a phone as you’ve not bought her one, the easy solution is to get her one.

You're still not getting the central point here OP - the other mum doesn't need her child to be supervised. If you do, that's on you. You could come later, you're the one that won't. You really are being incredibly unreasonable here - CF territory in fact.

Ah fair enough. Definitely something to work towards, it's so limiting not to (I speak as someone who didn't for a long time).

What about the phone, is that beyond reach? Then she could walk with her friend to the park and you could meet her there an hour later or so. Would free up some time for you too?

Why would she watch them when they don't need to be watched? Or at least, she doesn't feel her DD needs to be watched.

The previous arrangement was quid pro quo. The new one you’re proposing is entirely for your benefit. She’s not asked you to supervise her child, as her child doesn’t need supervision at all. You’re not putting yourself out for her, or doing her any favors.

You’re the only one here that wants their child supervised, and you’re choosing to do it for your daughter. That’s on you, nothing to do with the other mother.

Her daughter doesn’t need watching. You want her to put herself out to watch your daughter.

Is it just the 1 bus? If so, it’s time for her to take the bus home herself and for you to give her a brick phone. It’s completely normal for a year 6 child to get themselves to and from school.