Should there be restrictions on who is allowed to become parents?

With a tiny handful of exceptions for those who have demonstrated a lack of caring for others (psychopaths, convicted serial child rapists), no, definitely not. Instead there should be a highly effective support structure in place for both parents and children, including health care, mental health, education, and social safety nets.

Yes. Make everyone take a class, pass a test and get a license. Too many children are abused or raised improperly. It's a moral imperative.