I agree. He needs to check the voltage at the PARK LP fuse (20 amp) socket. That way he can rule out battery supply (hot at all times) and wiring leading up to. The Regal FSM wiring shows no connection with the multifunction switch based on his symptoms from what I can see. Headlamp control switch yes. The same switch provides the 12 volts to the IP lamps and dimmer control for the Instrument Panel and climate control, radio and center console background lighting. The taillights are part of the park light circuits.
The PARK LP fuse is located in the engine compartment fuse box.
Also, the original poster needs to clarify whether it is just an IP background lighting problem or are all lamps such as warning/idiot out at ignition to on.