What is the plural possessive of child?
First, ask what is the plural, because it is the plural that possesses, not the singlular child.
The plural is obviously "children"
Now apply the rule of forming a possessive: Add an apostrophe and an S.
The answer is chidren's.
It's the children's playground.
Also men's room, women's rights.
And if the Smith family lives over there, it's the Smiths' house because you don't add an extra "s" unless you pronounce it. You don't say "the Smiths-es house" - you pronouce it as "Smiths."
But you weren't asking about the Smiths, now, were you!