Well, I think similar topics have come up on the forum before, and we've never had a consensus about when it's okay to interfere with parental rights.
Taking a Rothbardian/Blockian view of libertarian law, I'd say that on the one hand parents don't exactly
have "parental rights," but then again one cannot forcibly "terminate" them.
It's a sticky subject, and not completely solved, but basically speaking parents aren't authorized to initiate aggression against children--for example, to assault or rape them. Children aren't property or chattel. Parental "authority" is a sort of "custodial easement" giving them limited rights. The child theoretically has the right to assert self-ownership, with the proviso that the parent can refuse to provide food, clothing and shelter. In other words, "As long as you're under my roof, you follow my rules." But custody, being a form of property right, can theoretically be bought and sold.
What makes the whole discussion infinitely thorny is that babies start out as non-sentient creatures, and libertarian law doesn't really know what to do with babies, vegetables or the senile. So everything I just said at most kinda sorta
works, and the fact remains that I can and do
resort to physical coercion over my young child when I deem it appropriate. At age 8 he now has a lot of negotiating power, but at age 3 he was a semi-human creature with very limited self-determination.