Could be the BTSI solenoid in the column or, more likely, the brake switch assembly. There are three separate contact switches in the brake switch assembly, normally without the brakes depressed the BTSI switch would be closed and applying power to the solenoid while the trans is in park. The brake switches like to break over the years.
As an aside, check fuse #16 in the dash fuse panel. Of course, you'd be missing your turn signals too if it's blown.
One last note: the BTSI solenoid is ONLY active with the ignition in run. If you just turn the key enough to release the column but not all the way to run, you will be able to shift without the brake pedal. That's just how these trucks are wired.