Do children get herpes from their own parents?

YES, from a kiss: Herpes simplex virus can be shed even when there is no cold sore visible. A simple kiss is the most common mode of transmission.

Depends: Genital herpes can pass during labor/delivery to a newborn but this often results in severe disease or death. If a kid got genital herpes it would be a finding of sexual abuse. Kids could get a cold sore if the parent passed this by kissing them while shedding the germ.