Your topic heading is a bit misleading, your query is not related to your parental rights but your parental and caring responsibilities. The legislation relates to unexpected and emergency situations when you are caring for someone.
Your employer is obliged to give you time off to deal with this, but not necessarily obliged to pay you while you do it, but they certainly can't sack you because you have taken time out for an emergency or a medical appointment.
Talk to your union about your situation and get some advice.