Yes it is on the parking brake itself.
The condition you describe is normal, check your owners manual on page 8 ( as far as when turning the ign to the ON position without starting ):
Brake system warning
Momentarily illuminates when the ignition is turned to the ON position, the engine is off and the parking brake is engaged. If the brake warning lamp does not illuminate at this time, seek service immediately. Illumination after releasing the parking brake indicates low brake fluid level and the brake system should be inspected immediately.
I would say the braking brake switch is screwed.
Also, as a side note, was your truck on the list of the recall for the Cruise Control Secondary Safety switch recall, and did you have it done ?