Can you give up your rights for a child if both parents agree?

No! There is this myth that a parent can just give up their rights. But its just a myth.

Only a court can terminate parental rights and courts are very reluctant to do so. There is a sticky note at the top of this forum that explains this. Just click on the family law link.

No, it is a myth that a person can give away or sign over their rights. And except for adoption normally the party who loses their rights still have to pay child support, ( not asked but often a issue)