When we saw an attorney in 2002 because XH was 2 yrs behind on support and was threatening us for custody.... we found out a few things.... I'll share what I know....
A MAN (and I use that turn loosely) can terminate his parental rights to avoid FUTURE child support payments but cannot do so until ALL ARREARS have been paid! (mine wanted to skip out on almost $15,000 but the lawyer told him he would be laughed out of the court and right into JAIL if he pursued custody changes!!!)
As far as not seeing the kids and having his parental rights terminated on that basis.... thats illegal (no judge in his right mind would do that).... (mine hadn't seen them in 2 years and was threatening to take the kids to avoid paying me support) Lawyer said that parental rights termination is rare and based on severe abuse, danger or etc... However, avoiding them, lying to them on the phone, not seeing them in years does not meet the criteria of their rights being terminated....
So while my XH has parental rights, he has not seen them in almost 7 years... his choice and he hasn't spoken to them in over 5 yrs...... he has told many people that he thinks they were mistakes.... (I'm so glad I got custody!!) What a guy! Best thing was that after our 2nd son, he got a vastectomy!! After our divorce, I felt a twinge of regret that he would never be able to have more kids, but after I saw how STUPID he really was regarding our sons, I am SO VERY GLAD NO OTHER children will ever have to call him DADDY!!