no you should not have to do that,when pressing the brake pedal can you here any type of noise from the shifter area,and there should be a spot on your shifter console to insert something,this is a park release override
DOES THE BRAKE LIGHT WORK
If so, Ok,now what you need to do is get your hands on a manual for this truck because you have to start checking the circuits for problem it easy just to replace parts if you have the money to throw away,you have to check to see if that switch is getting power and or ground to the release solenoid,we already know the switch is getting power because the brake lights work,but it could still be that switch,the solenoid release part of it could still be bad.
take a flashlight and look at the metal arm attached to your brake pedal,maybe 3/4 way up there will be switch with 1 to 2 plugs in it one of them is torque converter lockup release and the other one is the brake light/shifter release this actuates the park release solenoid
This is why you will need a manual for the wiring pictures and colors