What is the best way to sign over parental rights?

There is no "best way". The parent wishing to give up his or her parental rights must file a voluntary relinquishment of parental rights in the appropriate state court in the county in which they reside. The judge will decide if the petition will be granted and if so to what extent. A TPR is not a legal method that can be used by a parent to allow them to escape their financial responsibility to a minor child. Voluntary TPR's are usually granted to allow a minor child/children to become eligible for adoption.