"Once you do that it becomes your responsibility to care for and raise that person"
Until when?
I'm not sure if I disagree with you or not, but the post you are responding to isn't saying parents have no responsibilities to their children, only that a 13 year old could take on a whole lot more responsibility than is typical in our society.
So, maybe "get a job and take care of yourself" is too harsh, but is there some middle position?
Because she birthed him. Look, if a 13 year old kid wants to get a job, more power to them. However, if they don't, it's the parents' responsibility to provide for that child until they're an adult mainly because, at least in this society, kids aren't raised to be factory workers by 13.
You're looking at it from a perhaps pragmatic point of view, but you're missing the human factor; children aren't some investment you can liquidate if it doesn't work out. Having a child is a conscious decision to bring another human being into this world -- a human being who never actually asked to be brought. Once you do that it becomes your responsibility to care for and raise that person (or find someone else who can) regardless of whether or not they provide a return on investment.