One of the important logical consequences of the causal principle of parental obligations is that both the mother and father are jointly and severally liable for parental obligations to the child. Parents have obligations because they placed their children in a state of peril by causing them to exist. This creates a strict liability tort. The children have a legitimate claim on the parents to remove them from peril, which is done by raising the child to self-sufficiency in adulthood.
The action that creates parental obligation is making one's gametes available for fusion. If a man or woman engages in consensual sex or voluntarily makes gametes available for any artificial procreative procedure, then he or she has responsibility for any resultant child. This is because one has responsibility for the reasonably foreseeable consequences of one's actions, regardless of whether those consequences were desired or not.
It is hugely significant that the causal principle is the only theory of parental obligation which applies equally to mothers and fathers. All competing theories fail to obligate fathers. Since most libertarians who have written on the subject of parental obligations do not accept the causal principle of parental obligations, it is not surprising that fathers have been invisible in libertarian writing.
Rothbard's libertarian theory of parenting- I call it a theory of parenting as charity- argues that parents do not have obligations. For obvious reasons, this theory can never hold a man responsible for any children he begets. Whether he takes any responsibility for his children is his arbitrary choice. This theory of ethics also holds that the mother has no obligations, but since biological reality is that the woman gets pregnant, the effective result of implementing this ethical system in practice would be that a woman always faces the risk that she will be left to raise the child alone, since men have zero obligation.
It is easy to argue neither parent has obligations, but the fact is that a father can walk away from his unborn child whereas a pregnant mother cannot. The net effect of Rothbard's theory is to legitimise the despicable behaviour of those men who create children and then refuse to take responsibility for them, leaving the mother with all responsibility. Rothbard's theory that parents do not have positive obligations has blatantly sexist practical implications, albeit unintentionally.
The advocates of this theory do not want the theory to have such radically different implications for men and women. This is why all advocates of this theory are staunchly pro-choice (as Rothbard was). The appeal of abortion for this theory is that it addresses the practical barrier women face in declining parental obligations compared to men. Abortion supposedly offers women the chance to remove themselves from obligation, like the men can.
But this is all false. Firstly, you cannot legitimately give up parental obligations and the men are at fault for trying to do so. Secondly, killing the baby is not a legitimate choice (this is itself another logical implication of the causal principle of parental obligations). Effectively Rothbard's theory legitimises men walking away from their responsibilities as fathers, and for women it legitimises the fallback option of committing murder. This theory has many other awful logical consequences, such as the idea that child abandonment, starvation, and any kind of negligence are legitimate.
Some libertarians such as Roderick Long and Steve Horwitz advocate an alternative theory of parental obligations, but this alternative theory also has no way to obligate fathers. They argue that parents voluntarily accept parental obligations by agreeing to a tacit contract with society at large. The obvious implication of this theory is that only the parent who agrees to the contract is obliged. This logically means that the father is never obliged unless he chooses to be. Most advocates of this theory refuse to accept this clear logical consequence.
A variant of this theory argued by Roderick Long is that a father may be obligated to his child as a result of having played a role in creating the child, but the father's obligation is entirely dependent on the choices of the mother. What started out looking like an acceptance of the principle that one is responsible for the consequences of one's actions ultimately collapses into advocating that the arbitrary decree of the mother is the determinant of a father's parental obligations. The motivation behind this reasoning is to leave women the chance to remove parental obligations by abortion but prevent men from declining obligations if the woman does choose to have the baby. Whereas Rothbard's theory had sexist implications against women, Roderick Long chooses sexism against men. According to the logic of Long's double-standard theory, there is no reason why a woman could not go one further and both deny parental obligations for herself while simultaneously stipulating obligations for the father.
The idea that fathers can legitimately create children without having any responsibility towards them is absurd. Very few people actually believe this. Libertarian theory should accurately reflect that both parents have obligations to their children.