That’s the responsibility of a merit badge counselor and only individuals who are registered and approved by the council as a merit badge counselor can sign off on merit badges. Even unit leaders such as the Scoutmaster must be registered as a merit badge counselor and approved by the local council for any merit badges they wish to counsel. They just can’t do it because of the fact that they’re a leader in the unit.
And that means once a Scout finishes the Family Life merit badge, he can’t turn to Mom or Dad for a signature. Unless, of course, the parent is a registered and approved merit badge counselor."
Link: Merit Badge Sign-Off
More authoritative guidance can be found here: Scouting.org The Merit Badge Program