No. The rule is simple:
- For regular plurals ending in
s, the possessive suffix is
- In all other cases, the suffix is
Children is an irregular plural, not a regular one. Therefore, the suffix is
's, and the possessive form is children's. Your form *children's' is incorrect.
However, the other possessive forms are fine. Parents is a regular plural, so the possessive suffix is simply
'. That means parents' is correct. And you correctly write children's later in the same sentence. (Was your earlier mistake a typo?)
In this answer, the * symbol means that the spelling is considered incorrect.