Here's a couple of pics...
It looks like you can disable the Auto Parking Brake when transmission is in Park. However I don't see an option to disable Auto Parking Brake when in Drive or Reverse gears.
You left your truck on turned on when towing it?
Generally you would want to put it into neutral imo.
I believe putting the truck into tow\haul mode disables the auto park while in gear, at least it does for me. If I don't toggle it if my door is open it puts the truck into park from drive\reverse.
I will admit it is one of the more stupid and dangerous "safety features" of the truck.
Sounds like you have 2 differnet problems, transmission brake (park) and auto parking brake (parking brake) are 2 completely differnet things.
Looks like you had your truck thrown in "park" with the transmission brake. The parking break even on a new truck is unlikely to be able to hold the truck in motion (not 100% sure on these trucks)