Nobody chooses to be born. Does that mean that there are no moral obligations that arise by virtue of one’s existence as a human? Can obligations only arise from voluntary choice?

“Choice” is not the be-all end-all of morality (or, really, even particularly important). The obligation for parents to care for their children and vice versa arises from the nature of people and the nature of the relationship. Parents have an obligation to care for children even when they didn’t choose to have children. Conversely, children have an obligation to care for parents even though they didn’t choose to be born.