ok that sounds like a normal operation except for the fact that it moves, so yes it sounds like the parking brake isn't working correctly.

The parking brake has to be engaged for it to start, but it also has to hold the truck (not roll), and if i'm not mistaken the transmission shouldn't engage until you release the brake.

So check the wiring at the park brake switchs and check for any issues.
There are 2 switches on the monotrol pedal trucks, the switch on the right hand side of the park brake bracket which energizes the seat warning circuit when handbrake is released, it also puts the transmission in neutral by de-energising the directional solenoids.
The switch on the left hand side of the park brake bracket prevents the engine from starting unless the park brake is applied.