Can adults sue their parents for unpaid child support?

No. Child support is owed to the custodial parent.

The right to collect child support belongs to the custodial parent, not the child, even after the child is an adult they cannot bring a suit for back child support.

An adult child may be able to bring a suit for child support arrears on behalf of their parent's estate if they are the court appointed representative of the estate and any statute of limitations has not expired. In that case thy should consult with an attorney.