The question of children's rights and familial authority is often regarded as a grey area for libertarians, as it remains an issue of contention. I generally take a fairly anti-authoritarian view on the matter. While I think that Murray Rothbard's views on children's rights that he expressed in The Ethics of Liberty is an improvement over a more traditional conservative view, I ultimately do not find it to be entirely sufficient. In this regard, I genuinely think that Stefan Molyneux has provided a more rational libertarian view on children's rights and familial authority than Rothbard and this is his most significant contribution to libertarianism, although my own view is not identical to his.
On a normative ethical level I contend that the non-aggression principle applies to children just as much as it applies to adults and on a psychological level I contend that the imposition of any kind of physical violence is not necessary to raise a healthy child. I do not think that the consistent application of the non-aggression principle to children should be controversial, but apparently it is controversial, especially among many of the more culturally conservative libertarians. I see no reason why child abuse should be considered any more legitimate than adult abuse. That being said, I wouldn't necessarily want to blur the lines between a few light spankings and something more overt and egregious. But I still nonetheless would contend that spankings are not necessary to raise a healthy child.
Furthermore, for some of the exact same reasons behind why I oppose the state, I do not think that the mere fact that a child lives in their parent's household or the mere fact that they have a biological connection to their parents that this grants the parents the right to initiate violence and have completely arbitrary authority over every single aspect of their lives, nor does it mean that the child has an unchosen positive obligation to their parents. Even the capability of the child to run away is not a sufficient justification for whatever their parents do to them, and it is at this point that Rothbard's expressed views on children's rights starts to fail, since the love it or leave it argument is no more legitimate for parental authority than it is for a state.
In Rothbard's view, the child gains their rights as soon as they express the capability to run away. In my view, the child already has rights, it's just that their circumstances limit their ability to express them, particularly because of their dependance on their parents. This dependance is more understandable the earlier in childhood it is, but in either case it does not mean that the child has no rights. I do not think that children are the de facto slaves of their parents until they move out or get a job. In my view, parents are not owners of their children so much as caretakers. In a normative ethical sense, the child cannot be owned by anyone. No one can be.
I think that families should be voluntary. The fact of the matter is that not all families are voluntary, which is part of why a conservative view on the family doesn't make sense, since it broadly assumes the benevolence of "the family" as such. But I think that it is just as ridiculous to be "pro-family" as an absolute as it would to be "anti-family" as an absolute. The context that is missing from both absolutes is the actual behaviour of the family members and the consequential way in which the family is structured. A family can be generally healthy or abusive and parental authority could be nurturing or arbitrary.
I see no more reason to treat parents or family members as having intrinsic authority than to treat nations, states or corporations as having intrinsic authority. I don't believe in intrinsic authority or intrinsic value of any kind. I think that a transcendental concept of the family is just as irrational as a transcendental concept of society. Parents and family members should be judged as individuals and associate freely. An individual should always have the choice to disassociate with parents or family, as there is no intrinsic obligation. Otherwise, the family can be structured as a form of slavery, which sets up the basis for the authoritarian tribe when blown up on a somewhat larger scale and devolved.
The family, when it is voluntary, is the simplest anarchistic form of government, and it definitely deserves praise in such a context. However, when the family functions as an authoritarian institution, it is precisely what plants the seeds for the more large-scale forms of authoritarianism such as the state that libertarians commonly criticise. The initial breach of liberty always starts small-scale, at the level of the family and the immediately surrounding community. The logical and historical outgrowth of an authoritarian family structure is the authoritarian tribal system and monarchy. It is not a mere coincidence that monarchies are based on familial lines, and a tribe is essentially just a large extended family.
Another interesting point to consider is that in a sense authoritarian political ideology could be thought of as viewing political institutions as a surrogate family, so there is an important psychological element to all of this. While this tendency may not always be completely overt, it is nonetheless a fairly obvious connection. People may tend to want the state to play a paternal or maternal role because they feel that either they themselves or others in society are missing or in need of such a role or out of a feeling of obligation that can be traced back to a familial root. Likewise, the power-mongering of various individuals can often be traced back to a familial root. As long as one doesn't dive head first into Freudian absurdity, I think such an analysis can make a lot of sense and be very useful.