A child must be born before parental rights can be terminated.
You can seek to be awarded sole decision making with no specific provision for the father's parenting time and still collect child support, but if you consent to the termination of his parental rights, he will also have no more responsibility for the children. It is likely that elimination of child support is what he is actually hoping for.
If you allow the father of your children to terminate his parental rights through the court, then he will no longer owe child support. There is no way that the father can accomplish this over your objection.
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It is unlikely that a court will allow a parent to sign over his/her rights. The child must be born and then you need to get sole decision making and establish child support. Most parents, when faced with a child support obligation, will not completely walk away from their children. It is probably worth your while to meet with a family law attorney to gain a better understanding of your rights.
Woah, you really need to slow down. Yes, the baby has to be born before there can be any "voluntary relinquishment" of parental rights. If there is a "voluntary relinquishment" of rights, the father is not recognized as the father anymore, so he has no parenting time rights nor does he have any child support obligations. However, this is a long and difficult process and needs to be approved by a judge. As a former judge, I can tell you that I do not know of a single judge who will allow for a "voluntary relinquishment" without someone stepping up to adopt the child as their child immediately after the relinquishment. Both the relinquishing parent and the adopting parent need to go through extensive counseling and advisement of their rights through the Department of Social Services. So, this is no "quick fix" and you need to think long and hard about this and the fact that what your proposing is almost certain to be denied by the Court. Judges want fathers to be financially accountable for their kids. Judges also want children to grow up with both a father and a mother. If the father refused to "step up" and assume responsibility to be a part of the kids' lives that "is on him." No judge is going to let him off that easy. He is going to at least be forced to "man up" to financially provide for the children. He produced the children and it is not fair to have taxpayers bear the burden of supporting the children.
The information provided in this answer does not create an attorney-client relationship. If you are interested in his legal services, feel free to call Chris at (303) 409-7635 at his law office in the Denver Tech Center. All initial consultations are free of charge.