Well my own view is that parents do have positive obligations to their children. I got this by combining what Block says here about abandonment with what Kinsella says here about how children become self-owners. In that paper, Block says that parents do have one obligation at least: they cannot abandon their child without informing anyone, and if they do want to abandon their child, they must make it available for homesteading (i.e. available for some other person/couple to raise the child if they want to).
All property owners have ‘positive obligations’ of some sort. Namely, the property owner must do whatever is considered necessary to constitute non-abandonment of that property. Take the example of a field. A property owner cannot simply leave the field unused indefinitely. At some point (say after X years), the field must be considered abandoned, such that someone else can homestead it and become the new owner. If the previous owner returns after this, the libertarian court would not rule in his favor, because they would say that what he did (i.e. leave the field unused for X years) constitutes abandonment, and so the new owner did nothing wrong. So we can say that if you own a field, and you want to continue owning it, you must: not leave it unused for X years, i.e. do whatever is considered necessary to constitute non-abandonment.
And so it is with children. There are certain actions that constitute abandonment, and if a parent abandons their child, the child becomes available for homesteading, i.e. someone else (the courts will decide who) is justified in assuming the parental rights over the child. So if you see someone else letting their child die of hunger, there is a course of action you can take to help that situation… a libertarian court may consider the parents’ actions to constitute abandonment. (See this thread for more about this).
In this sense, a parent has positive obligations to their child: they must do whatever is considered necessary to constitute non-abandonment. I like this way of looking at things because it doesn’t involve making ownership of parental rights a ‘special case’ of property rights, and it gives us a coherent libertarian answer to the difficult issues of both child abuse/neglect and abortion.
Rothbard and Block both baulk at the idea of ‘positive obligations’ (because it usually means something quite different), but this does cause them to say things that appear contradictory, like the one you pointed out.