But even in that stupid, dehumanizing framework it still ought to be one of the issues of "parents rights" they love so much. Your child's privacy being violated is a violation of your property rights. YOU didn't consent to that child's privacy being compromised, and they are a thing that belongs to you and can only exist according to your beliefs and rules, so that was an attack on you.
So the real truth is that to conservatives, there is no coherent ethical framework they can turn to to reliably make judgements. It is the politics of being a cruel and obstinate asshole.