a place to "leave" / "park" a child

I wouldn't recommend it in this context. By using park you are (metaphorically) seeing the child as an object, like a car - ie something just to be left and forgotten about till you need it again. It can be used informally and a bit ironically ...

I have to go shopping, so I'm going to park the kids with my Mum....

but in a context like yours, it makes it sound as if the parents don't care about the children.

Thank you, Sue.