You don’t get it. Marriage is a common law contract. Parental rights come from that and are based on that legal tradition.
If you do away with that tradition and there is no marriage, then there are no parental rights left. The state will make new laws regarding children and they won’t be parent friendly. Ever read the UN Rights of Children?
Don’t think for a minute they will base rights on blood kinship, they won’t, they’ve been denying men their biological parental rights for years now, it’s only a short step to denying women’s also and predicating it all on the “best interests of the child” and “public policy interests”.
If the state has such sinister intentions then will state-sanctioned marriage really serve as an effective bulwark against their designs?