If the parents were not married at the time of the child’s birth and the parents subsequently married – can the child be re-registered as if they were married at the time of birth?

Yes, If the parents were not married at the time of the child’s birth but they could have been (no legal impediment – married to another or underage) and they subsequently got married then they may do a re-registration.  The registration would now appear that they were married at the time of the child’s birth.  The original registration is marked “re-registered” and is not accessible except by Court Order.