If you are the control operator at the station of another amateur radio operator who has more certificate qualifications than you, what operating privileges are you allowed?

As a control operator, you are legally restricted by your own personal certification, regardless of whose station you are using. You cannot "borrow" the privileges of the station owner just because their equipment is capable of it or because they hold a higher certification. The golden rule of amateur radio privileges is that you can never exceed the operating privileges granted by your own certificate. If you only hold a Basic qualification, you are strictly limited to Basic privileges even when sitting at an Advanced operator's station.