With the consent of both parents, I've seen it happen. But generally no, a parent can't sign over parental rights unless another parent--usually a step-parent--is willing to immediately adopt the child in place of the biological parent.
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Generally, no. This is because in order to no longer be responsible for child support, the father would have to either have to have a disestablishment of paternity proceeding (which would require that he be proven not to be the biological father), through a termination of parental rights action pursuant to chapter 39 of the Florida Statutes, or your children would have to be adopted by a stepparent, which would terminate the biological father's status as "legal father." Absent any of these proceedings, the father cannot merely "sign his rights over."
You can, however, stipulate to having sole parental responsibility predicated on the fact that it would detriment the children for you and the father to have shared parental responsibility. You can also stipulate to a downward modification of the father's child support obligation to a point where it's nominal or zero, so long as there is a specific finding (by the Court) that it is in the best interests of the children to deviate from the father's statutory obligation. This type of agreement might satisfy both of your desires, so long as a Court finds that it is enforceable and complies with all statutory criteria, making specific findings where necessary.
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I agree with Attorney Hopkins and would add that it may be difficult to have the child support reduced if the case was initiated by the State Attorney's Office or Department of Revenue because they would retain an interest in the case for the purposes of child support and would likely oppose any attempt to lower the child support without good cause (i.e.not just because the father doesn't want to pay or because its to bothersome for you to continue to enforce the child support order). Child support is the right of the child not the parent and the Courts are very reluctant to interfere with that right because of disputes of the adults.