One of the most important philosophical questions relating to the family is whether parents have enforceable positive obligations towards their children. How you answer this question depends on your theory of the relationship between parents and children. Here are 3 major theories of that relationship:
The Theory of Parental Ownership
The Theory of Parenting as Charity
The Theory of Parental Responsibility
The first two theories deny parental positive obligations. The third affirms them.
The Theory of Parental Ownership
The theory of parental ownership views parenting as a kind of property ownership. In this view, parents are the owners of their children and children are the property of their parents. This is an ancient theory. Versions of it have been expressed by Aristotle, codified in Roman law, and argued by Hobbes. This is the main line of argument:
Parents create a child
Creators rightfully own what they create
Therefore parents rightfully own their children
Owners can do whatever they want with their property
Therefore parents don't have any positive obligations
Sometimes this theory is modified to say that parents may have obligations towards other adults that indirectly influence their treatment of their own children. For example, some posit that a parent must inform other adults before abandoning a child in case other adults would like to adopt. Another example is the idea that parents must ensure that they do not raise children who will aggress against other people. Yet these kinds of obligations are ultimately not to a parent's own children but rather to other adults.
The Theory of Parenting as Charity
The theory of parenting as charity denies that children can be owned. It views children as self owners who have rights. According to this view, parents do have some negative obligations: they must not to actively abuse (aggress against) their children. These are the same general negative obligations that any adult has towards another adult. However, according to this theory, parents do not have any positive obligation to provide care to their children if they don't want to (or if they stop wanting to). The theory of parenting as charity has this main line of argument:
Parents create children
Children are temporarily helpless
Creation itself is not a source of obligation
Helplessness itself is not a source of obligation
Therefore there are no enforceable parental obligations (making any parental care a voluntary act of charity).
Why isn't creation a source of parental obligation? One argument given is that if this were the case, parents would be obliged to care for their children forever and this is taken as self-evidently false. Why isn't helplessness a source of obligation? The argument is that the mere fact that some person is in need does not logically create an enforceable obligation on any other person to provide for them.
The Theory of Parental Responsibility
The theory of parental responsibility also denies that children can be owned. It differs from the theory of parenting as charity in that it argues that obligations can come from a different basis than merely the act of creation or the fact of a child's helplessness. The core argument is as follows:
People are responsible for the consequences of their actions
As a consequence of creating a child, parents have put another human being (the child) in a state of peril.
Children cannot consent to being born
Since the child did nothing to create their state of peril nor consent to it, the child's peril is entirely the responsibility of the parents
Therefore parents have a positive obligation to do whatever is necessary to remove the child from a state of peril, since not doing so would constitute an act of aggression as a form of tort.
The Libertarian Error on Parental Obligations
Most postwar libertarians who addressed this question have denied the validity of enforceable parental positive obligations. For example, Murray Rothbard, Wendy McElroy, Walter Block, and Harry Browne all denied them. Libertarians usually justify denying positive parental obligations by arguing some variant of the theory of parenting as charity. Rothbard provided a detailed defence of this idea. He saw the needs of children as just a subset of the more "general problem of the sick or the helpless". Although Rothbard never used the term charity in relation to this, I think it is a logical name for this theory.
I believe that the denial of parental positive obligations is a huge error in libertarian theory. Only the theory of parental responsibility (which affirms enforceable parental obligations) is logically defensible. A few libertarian thinkers have argued for variants of this theory in the past. Nathaniel Branden, Doris Gordon, and John Walker all made arguments along these lines, but only in short fragments. In present times, Stephan Kinsella is the main proponent of this idea, although it is not the focus of his work. I believe parental responsibility is the correct theory and the only one that is ultimately compatible with the broader principles of libertarian philosophy. I'm currently researching this subject and will be addressing these arguments in more detail in future.