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.
I agree that their logic isn't consistent, but in this case I would say their solution isn't to improve the privacy of their childrens' porn access, their solution is to lock down their childrens' behaviour so they cannot ever see porn. They're not imagining their children as complex beings in this instance, they are objects.
The suggestion that online privacy when accessing porn is something that will affect their children would sound like an admission that you want to show their children porn. If you point out that their kids are going to find porn on their own because that's just how the world works, they won't investigate that. They'll just fall back on their overdeveloped disgust reflex and attack you for it.