I guess we will have to agree to disagree on this, because I am almost dead certain that while I was going through my divorce and custody issues in Arizona, my attorney told me that two parents could not agree to allow either one to stop supporting the child. I really don't like to discuss my personal business here, but I would have gladly agreed to allow my ex off the hook for child support if that meant he would have given up his parental rights, and the ex would have gladly not paid a dime of child support if he could have avoided it. However, the law would not allow him to sign off, even though I was more than willing to allow him to do so.
It is different if there is an involuntary termination, as in child abuse/abandonment, but I am pretty sure I remember my attorney telling me that the ex could only sign off on his rights if I remarried and my new husband was willing to adopt the kids.