It seems that based on what was proposed in the OP, for the purposes of this discussion the key IS parenthood responsibilities. If a man does not want the responsibility and conception has already occured, should he be able to decide for himself to NOT have any parental responsibility just as the woman who gets pregnant then decides to abort the pregnancy gets to make that very decision?
On the surface, I want to say "hell yeah, a man should be able to choose!" But as I consider the bigger picture that goes beyond the two people that spawn the child, I'm forced to pump the brakes a bit. The fact of the matter is that the child that comes into the world is very likely to have what I'd call a societal impact. It's generally been up to the parents/families of children to prep the children for society and to take on the responsibility of the child's well being so that the parties that have no affiliation with the "sexing duo" that make a baby won't have to shoulder a responsibility for childcare, etc.
Another consideration is what I'd refer to as the human factor. What if the man initially doesn't want the responsibility, but later has a change of heart and decides to enter the picture somehow? I don't see how this can be effectively legislated against and think of the issues it would present when it inevitably occurs. I doubt that forcing the father to pay back child support would be a suffient recource and in most situations I doubt that that would be any more feasible than expecting everyone and their children who is in this country illegally to go back home and then try to re-enter through the current citizenship process.
So although it's real messed up that the man gets less of a say so (really no say so), that's life. Ain't much equitable able the fetal incubation period where momma gets to experience the pain, discomfort, and hormonal reprocussions of being pregnant while men get to continue to go about our normal lives. So again, that's life and life was never promised to be fair and equitable.