If the brake lights are not working, AND the vehicle will not shift from Park, then the most likely culprti, is in fact, the Brake Light Switch (aka Brake Pedal Position Sensor). This switch is located at the top of the brake pedal arm. Make sure all the fuses are good, and if they are, the only common factor of no brake lights and no Park release, is the brake light switch.
If it was the Shift Interlock Solenoid, then the brake lights WOULD work. The solenoid has no bearing on the brake lights. I would double check all the fuses, and check that there are no loose, disconnected or broken wires at the harness for the switch. If the switch works when you use a srewdriver, then it just may be installed incorrectly. Like I say, the only common factor here is that switch.
Just doing a little digging... on this vehicle, there is also a Brake Light relay in the underhood relay box. Swap it out with a similar relay, and see if that helps. If so, replace the relay.