it means "read' or "go through"

here “thumb through” is a phrase it means to turn the pages of a book quickly in order to get a general idea of what is in it.

Hello hiyaya, 'thumb' is a finger and 'Thumb through' is an expression, which means to quickly look at a pile of papers ,here through their children's diaries without reading it carefully .

it could also mean that the parents were reading the child's diary without their knowledge