Your problem makes no sense. Even if the mum wanted to help, she couldn't. If the girls only decide moments before going to the park, how could the other mum arrange with you that you can stay home and she will take a shift? You would already be there anyway. Equally the mum can't say "I'll watch them on Wednesday" because what if it starts to rain on Wednesday at 3.15 and nobody wants to go to the park? Should she ring you to say its cancelled and you charge across town while your daughter waits on the pavement for half an hour? Or are you also expecting the mum to host your daughter in the event of rain? That would mean she has to make sure the house is tidy and have snacks in. That's quite a lot to expect of someone who gets nothing from you except supervising her daughter at the park when she doesn't need or want it.
In fact, this issue has nothing to do with the other mum. Whether they go to the park or not, she would stay at home. The only reason she would have to leave her house is to collect your daughter from school, which really isn't her problem.
Can you get your daughter a permission slip to leave school without you and if you're there, good and if you're not, that's also fine.
You'll need to get her a phone sooner or later anyway, so you may as well get one now. The only problem is that she'll probably text you 2 minutes before she goes to the park, in which case, you'll be there anyway! But again, that's nothing to do with the other mum.
Is costing you a lot in time and money to pick her up. You make the journey to school 4 times a day. Maybe take her in the mornings but say to her that she can make her own way home. She'll know what to do as she's done it many times before. If she has a phone, she can update you on where she is and if you get something like Family Link, you can see where she is at all times anyway. She might really enjoy being 'grown up'.
Whatever you decide, none of this has anything to do with the other mum. It's unreasonable to expect her to stop whatever she's doing so she can watch your daughter in the park, and it is only your daughter, as her daughter doesn't need anything.