To feel this mum should take her turn?

Friends mum would be nearby in case if accident, or friend could call you? Or give your DD a brick phone?
I think there are lots of ways to solve this and you need to clarify to yourself why you are reluctant to do so.

Get your daughter a phone. It doesn’t have to be a smart phone, just something she can call you from. Then let her make her own way home. She has to get used to it at some point, unless you’re going to be one of the pathetic mums my DC see waiting for their secondary age kids at the gates.

You need to be teaching your daughter independence and planning for year 7. This situation is a great opportunity to start that. Do you know how she will get to school then?

You need to give permission for your DD to leave school without you and then pick her up later from the park or you need to start saying no. It's that simple.

Nothing to do with the other girls mum.

Tell your daughter she can't go to the park after school anymore. You don't want to find any other solutions so it's a blanket no.

Then get your DD a phone or start saying no. It's not difficult.

Then it sounds as though the real issue is that you’re not comfortable with your DD being unsupervised in a park and starting to have a degree of independence.

That’s your decision to make, of course, but you can’t expect other mums to compensate for that.

Either let your DD have a phone and trust that if something goes wrong, someone will call you, or suck it up and sit in the park, cold and bored.

If you have to pick your own DD up anyway or they won’t release her, how would the other mum supervise them? You’d be there too anyway.

Just fill in a permission slip for her to leave on her own and tell her/her friend to call friends mum in an emergency since she’s closer and then call you. It’ll only be for 45 minutes or so anyway if you’re already planning to come and get your DD from the park later on anyway!

Then the only solution is no park 🤷🏻‍♀️

I think you are potentially making it more difficult than it needs to be.

You can say "no" to your daughter if it's not convenient.

You can fill out a permission slip to allow her to leave with her friend - and only her friend - on certain days. Her friend obviously has permission to leave every day on her own, so it's not impossible.

I would be having a conversation with my child to say it's not convenient to go every time she asks, and in future her and her friend needs to decide a day or two in advance - or go on set days each week - so that you can sort out a permission slip.

As someone else has pointed out, the clocks go back in a little over six weeks so that will probably change things.

I had to travel when my kids were young so this happened me a lot.
But I chose the school and distance so sometimes you just have to go with the flow.
Take a book, head to the nearest shop but neither child needs supervision.

If it is such a problem say no park and go home

So you are saying again that your daughter's friend is not in need of supervision at all, due to her phone, where she lives, arrangements her family have put in place etc. You still think you daughter does still need supervision. But somehow or other you think is up to your daughter's friend's mum to be responsible for supervising your daughter, rather than you.

I think it’s fine to have a friendly chat with the Mum about this.

But I also think you can say no to your dd, or that park trips have to be prearranged so you know not to come out for her until later.

Sorry but it's not up to this other mum to supervise or pick up your daughter. She is allowing her daughter to do things on her own, so it is entirely your responsibility to take care of yours.

Can she not go to the park with her friend straight from school, and you collect her later? You say the children arrange it at school and you don't find out until you're at the gate - well guess what, if it's not possible, you say no!

Why can't she catch the bus for 20 minutes? She has a phone and can contact you.

I imagine that if you explain it to them they will, they're probably just not letting her go because they know you're coming. If you said she was going with her friend to play after school and she was allowed to walk with her, they surely would let her go.

Surely it’s nice for your DD to have a friend there to play with.

If you don’t want to go on certain days, just say no because you need to get home for x,y,z

If you are already there at pick up what do you expect the other mum to do? Surely you wouldn’t travel home then come back to collect her when they have finished playing?

How far is the park from school?

Could she go to the park and you meet her there or walk her to the park and then let her walk home?

"No love, not today"

That's all you need to say to your child on the days you don't fancy taking her to the park 🤷‍♂️

It has nothing to do with the other mum.

I’m sorry, I’m baffled by this.

Your year 6 daughter is not in charge of your life.

So if you don’t want to supervise her at the park (whether or not her friend will be there) you just say “no, not today” and head straight home.

Maybe explain to her in advance that she can’t just make social plans and announce them to you at the end of the school day anymore!

But the other mum is not doing anything wrong here. What you’re really asking her to do is to come out just to look after your daughter at the park as a favour to you. You can ask her, but she’s not obligated to provide you with childcare. It’s important to remember that you are not providing her with childcare, she’s happy for her daughter to walk around and go to the park by herself.

It's not the other mum's fault you live further from the park than they do. Get your DD a phone just for calls and texts doesn't have to be a smartphone if you're so bothered. She can contact you if they decide to go and agree on a time to collect her so you have no waiting around.

Fill in the consent form so that she does it have to wait for you to be there whether you have come to pick up or not.
Make it clear that plans need to be arranged in advance and If not planned a park trip then you go to get her and head home. If one is planned and plans have to change, can't she walk back to herel friends house and you collect here from there? (Friend or friends mum and let you know if you don't want your DD to have a phone.)