The Theory of Parental Responsibility holds that parents have enforceable positive obligations to their children because in creating a child the parents are responsible for placing the child in a state of peril.. Given that parental obligation only arises as a consequence of action, there are cases in which a child can come into existence without a parent incurring positive obligations.
Rape
A rape victim would not have parental obligations for any resultant child since the rape victim did not take action that resulted in a child being created. It was the rapist who acted, and thereby forced those consequences on her.
The rapist would have obligations to support the child, although a reasonable legal system would presumably deem him a criminally unfit parent and deny him access, so his obligations would have to be in the form of monetary payments to whoever was the caregiver of the child for the child's support, alongside restitution payments to the victim.
As a side note, the fact that there are no positive obligations in cases of rape victims means that parental obligations do not provide an argument against abortion in those cases. In cases of consensual sex, parental positive obligations do refute the right to abortion, but not in cases of rape. Whether or not abortion is justified in those cases depends on other arguments.
Gamete Theft
In cases of gamete theft, the biological parents would not have parental obligations. One qualification is that gamete owners have an obligation to be responsible owners, since gametes can have such enormous consequences on the lives of innocent others. Rivka Weinberg has argued that gamete owners have an obligation to take the same level of care with their gametes as one must do with a hazardous material.
Presuming that the biological parent did exercise due care, he or she would not incur parental obligations. For example, if a student donated sperm to a respectable university research team for the expressed purpose of aiding a scientific study of sperm motility and his sperm were stolen from the university and used instead to create a baby, he would not have parental obligations. He could not have reasonably expected to create a baby by his action.
Paternity Fraud
A man fraudulently told that he is the father of a child (when in fact he is not) does not have enforceable obligations towards a child, since he is not the one who created the child's peril. Believing a fraudulent obligation claim does not convert the fraudulent claim into a real obligation. The child's legitimate claim is against the natural father. Whether or not the duped man chooses to become an adoptive father is a matter of personal choice for him, but not obligation.
Endangered Life Of The Mother During Pregnancy
If the life of a pregnant mother is endangered by the unborn child, the mother would have a presumptive right of self-defence against the child to preserve her own life. Her self-defence right would nullify her parental obligations, assuming that the risks to her life were not the result of reckless negligence on her part.