A termination of parental rights does not terminate the duty of either parent to support the child before the child’s adoption unless that duty is specially terminated by order of the court after notice of a proposed termination or relinquishment is given to the department of human services in the manner appropriate for the service of process in a civil action in this state. A termination of a child support obligation under this section does not relieve a parent of the duty to pay any unpaid child support.