My son's father wants to sign over his parental right because he states that he can't spend time with his other son. Does he have the right to sign over his parental right because of his other son?

Any parent has the right to relinquish their parental rights. Doing so simply means that they give up any say in the rearing of the child and any custody or visitation. It does NOT release them from financial support obligations.

I think its wrong to relinquish one's rights except if its to allow someone else to adopt. All that it does is potentially harm the child.

Many men do it by just not spending time with them, they don't have to "sign any rights away" they just don't visit, don't spend any time with the child, you can not force them to visit, you can not force them to take the child on their days.

All you can force them to do is pay child support. ( he can not get out of doing that)

He doesn't have any "right" to "sign away his rights" for any reason. He can, of course, choose not to exercise his rights but that's not the same as "signing them away" or having them cease to exist.