It will shift from park with the switch turned toward the start position but before the engine cranks.

check your linkage. the vibration of you cranking it up might make the stick to move which it shouldnt because of the brake lock switch
keep your foot of the brake and start it like that and see what she does

Shift interlock solenoid not getting power from the brake light switch. Check to see if your brake lights work. If so, check for power to the solenoid.

l do not understand what is suppose to be the problem it should shift from park without the engine running. All that needs to happen is to have the key turned out of the lock postion and depress the brake pedal and it should come out of park. Maybe you should check how you worded your question.