Children and other legal dependents have only two natural rights: the (negative)
right to life, and the (positive)
right to emancipation. The former means that from the point when a baby becomes physically autonomous (i.e. birth) to the point of death killing a human being is a major no-no: regardless of age, mental capacity, dependency status, and so on. The latter means that the parents can't do anything to prevent the child from becoming a self-owning individual upon reaching an age of maturity, or to prevent them suing to be adopted or emancipated early, or to prevent someone else suing on the child's behalf. If a parent / guardian violates those rights, then use of violence to protect their children (i.e. by abducting them) is fully justified.
This means the parents have an obligation to disclose some information to the community at large: that they have a child, the age of that child, if that child is mentally handicapped, if they can't afford / don't intend to take care of that child so that it may die, and so on.
Aside from that - parents have the right to do whatever the hell they want. There are plenty of natural mechanisms that will discourage child abuse: ostracism, explicit contractual obligations to follow a code of conduct that are created through marriage / church / school / homeowner association / employment, and so on.
Yes, some parents will have children, not tell anybody, and keep them locked up in a basement for the duration of their short and miserable lives with no one knowing their rights are being violated. This happens from time to time in spite of all the government "child protection" bureaucracy that is already in place, and it will happen in a free society as well. This mere possibility does not create a "divine right" for a government entity to go around searching everyone's basement.
Yes, some parents can refuse private child welfare agencies / NGO's from checking up on a child, but those refusals will be tracked and after a while there'll be a serious cause for concern. Those agencies will be fully justified to assume the worst and publish blacklists of suspected child abusers, which is a strong encouragement for cooperation.
Yes, some parents will use "inappropriate" levels of violence or sexual contact with their children, but inappropriate according to whom? Child sexuality has been widely recognized in most cultures of the world before those cultures were affected by the prudish Abrahamic religions. Those subjective mores should be dealt with through public opinion -- reputation and ostracism -- not law! Hurting a child to the point of death or brain damage violates that child's natural rights, but that aside - the parents make the rules.
The issue of parents' rights isn't just a moral issue, but an essential pragmatic one as well. It has been clearly demonstrated that making parents the servants of their children, as has become the case in most industrialized countries throughout the world, leads to
unsustainably low birth rates, and thus tremendous (though initially invisible) economic loss. Who in their right mind wants to waste so much time and money, as well as physical and emotional energy, raising little snitches for the state?! As religious brainwashing gradually wears off, ever-fewer people choose to have children, and especially more than one, while a societal average of below ~2.2 kids per woman leads to a shrinking and aging population, thus fewer and less innovative workers, skyrocketing prices, falling IQ levels, and falling quality of life.