It should be up to a neutral arbiter, perhaps a social worker, not necessarily the doctor, if it gets to that point. And there needs to be a very sharp dividing line or we'll get nonsense like a doctor suing to have a child taken away from its parents because they feed it fast food too often.
If it's a cold, fine. If the kid has anything requiring medical care, I don't give a crap if the parents worship the great pumpkin...the kid should get help.
I hear you on losing the stupid genes, but I question the first statement. 'Rights' aren't absolute- they're balanced by responsibilities and other people's rights. So no, parents' rights should not always trump the government.
Parents have no right to treat their children as property rather than as people for whom they have a duty to care. If they don't want the responsibility of care, then they can give up parental rights by adoption or refuse to ever acquire such rights and responsibilities by birth control or abortion.