If a parent gives up his or her parental rights is he/she still obligated by law to make child support payments?

Giving up parental rights is different than one parent having residential custody. It is highly unlikely that the court will allow father to give up his parental rights because he does not want to pay child support. Such a drastic remedy is usually taken because one parent's conduct is harmful to the child, e.g., mother is addicted to drugs; father is an alcoholic.