How can I be charged with child support and not have rights as the father?

Child Support and Child Custody/Parental Rights are two separate legal issues. Child support is where that State has ordered you to pay a designated amount for the purposes of your child's well being based upon your current income and state guidelines. Child Custody/Parental Rights is where the court, generally family court, has awarded each parent certain legal custody, joint custody, or visitation rights concerning the children in question. For most states, when the parents of a child are not married or divorced, and one of the parents petitions the court to seek child support, the court will set a hearing date for both parents to determine if child support will be awarded, and if so, how much. Custody issues are where the court will address any and all custody issues concerning the children. This is sometimes done at the child support hearing or may be scheduled as a separate hearing. The question I have pertains to your meaning of "rights as a father" in your question. Not certain of the facts of your case, but you will always have parental rights as a father, unless your parental rights have been removed by the Courts. My advice is to contact a family law attorney in your area to discuss your situation in more detail to help you obtain the goals you seek through child custody.