One ancient theory of the parental role is that parents own their children as property. The vast majority of classical liberal and libertarian thinkers have been against this theory ever since Locke published the Two Treatises in 1689. Locke spent the whole of the first treatise arguing against Robert Filmer's Patriarcha. Whereas Filmer had argued for unlimited parental (patriarchal) ownership of children, Locke argued that parental authority has limits and children have rights.
Locke faced a major problem in opposing parental ownership that dogged libertarian thought for centuries: Locke’s own theory of homesteading— if applied to determine the ownership of bodies— logically implies that parents own their children. The homesteading rule states that the rightful owner of a resource is the creator or first user of it. In the case of children, parents are the creators. Locke tried to resist the implication of homesteading theory that parents own children but did not have good arguments. So why don't parents own their children?
The core premise of the theory of parental ownership— that homesteading is the method of establishing ownership over bodies— is incorrect. Homesteading is not the only way to allocate property rights. Property rights are rightfully assigned to the person with the best objective link. For unowned physical objects, the best objective link is indeed demonstrated by first-use homesteading.
However, when it comes to human bodies, each individual has the best objective link for ownership of their own body. This is the objective rule used for establishing self ownership of bodies. Babies own themselves because nobody else has a better objective claim to own them. The best objective link is also not dependent on the capabilities of the baby. It doesn't matter if babies cannot yet argue a syllogism or advocate for their rights, they still have the objectively better claim to be the owners of their own body simply because they are the inhabitants of that body.
The fact that self ownership is the best objective link is irrefutable. Anybody who wanted to claim that their parents (or anyone else for that matter) have a better claim to their own body than they do would be contradicting themselves in the very process of arguing. This is because one must accept self ownership in order to use one's own body to argue, so one cannot refute it.
Furthermore, the theory of parental ownership is logically flawed because there is no way to account for an owned child becoming a self-owning adult. You cannot acquire ownership of yourself because an individual cannot homestead anything unless he already has self ownership. So how is a baby supposed to acquire self ownership? Self ownership must be presumed from the get-go as the only non-contradictory basis for libertarian ethics. All other libertarian principles, such as homesteading, depend on the assumption of self-ownership. There is no way to come up with a non-contradictory account of an individual starting as property and becoming a self owner because one cannot homestead oneself.
In short, parents do not own their children because humans cannot be owned. Humans are self owners.