Im starting to very seriously think that one should not be able to have kids outside marriage or a marriage-like contract. The problem with rothbards idea of the parent-child relationship is that he fails to subscribe the parent to any kind of responsibility for the existence of the child and therefor sees the child more as some alien figure that the parent may wish to have company with, a relationship that can be terminated at any time as if it were similar to any other.
In fact, the parents must have ultimate responsibilities for the existence of the child and therefore also for its well being until such a time where is has assumed its full set of rights (adulthood). Hence, there must be an implicit parent-child contract that takes into consideration reasonable expectations and duties in raising and generally taking care for the child.
This contract must be based in reality, including historic, biological, social, and scientific, and must reject modern lies (reality denying) and pseudo-scientific conceptions. In practice, this means that a child must be raised in a stable mother-farther household where appropriate resources are given to both the general welfare of the household, and specifically to the child themselves. Economically, the most efficient and correct way of achieving this is through division of labour between a stay-at-home caretaker (mother) and industrious worker (father).
Marriage and marriage-like contracts ensure this happens and traditionally are very difficult to annul, carrying significant penalties and reparations for breech or withdrawal of contract.
In the case of those who do have a child outside such a contract, then they must be assumed to have entered into one on birth of the child, and breech of contract/duty must be able to be resolved in court by the other parent or on behalf of the child.