As far as I am understanding it, we are discussing men relinquishing ALL parental rights, and not paying child support.
That is different from simply being a non-custodial parent. A non-custodial parent still has legal rights to the child, and thus may pay child support if requested. They are still, oficially, a parent -- just not one the child is living with on a significant basis.
We're discussing a man who basically gives up his child for "adoption" (presumably to the woman, or perhaps to someone else if she doesn't want it either). He has relinquished all of his parental rights.
A woman who gives up all rights to a child does not pay child support.
What we're arguing about here is, basically, the fact that men are generally not allowed to give up parental rights.