The parent will transfer with the scout, once the scout is transferred, since the parent is connected to the scout. The parent’s pack role doesn’t have to end in Scoutbook unless that role is actually ending. To end the pack leadership position in Scoutbook, the parent can go into Scoutbook to My Dashboard -> My Account -> My Positions, click on her pack role, and add an end date.
To add a troop leadership position in Scoutbook, in your account go to My Dashboard -> My Units -> Unit Roster, scroll down to the bottom of the adult leaders list and click on “Add Leader”. Search by the parent’s email address (since she already has an account), and add ether new troop position to her account.
Obviously, there is paperwork to be done for n the BSA in order to add the adult to your unit as a leader and end her position with the pack. I assume you were only looking for the Scoutbook side of things.