Parental rights can not be "signed away/over" without a court ordering so. A court will not order so unless one of several conditions are met or exisit:
1) Divorce related---- NCP parent has abanonded the child for the Statute specified time both financially and physically AND there is someone to step in the NCP's place to adopt the child. A divorced parent can not say "Nope, I'm done, I'm not this child's parent anymore I'm signing away my rights" Doesn't happen. Ten Fifteen years down the road that parent will still be held for financial support of that child if the custodial parent goes after it.
2) Child In Need of Care -----(different statute wording for each state) child/ren have been removed from a dangerous/neglectfull home enviornment. Child becomes ward of the state, parent/s are given a specified amount of time to correct the home situation. If situation is corrected child/ren return to parent/s. If situation is not corrected children remain wards of the State, move to foster care, parental rights are terminated, and children hope for adoption.
3) Adoption (infant)----- pregnant mom opts to have child adopted. Child is born, biomom surrenders child to adoptive parents or agency. Hearing is held or without hearing mom agrees and parental rights are terminated.
Flat out that is it. In no way shape or form can a parent "sign over/away" rights to a child in any other LEGAL way. A parent can give GUARDIANSHIP to another person BUT if this hasn't been stamped by a Court of Law this guardianship isn't legal and will not stand without a hearing establishing custody.