My 2014 won't back up if door is ajar and you don't have your seat belt on. Maybe the cold weather is causing door button not to signal door is closed. Of course if you have seat belt on this should not be activated anyway.
WHAT???? If that ain't ridiculous I dunno what is.
This coming from someone who isn't crazy about seat belts and can't stand any "controls" that force me to put it on before my vehicle is allowed to do what I am trying to make it do.
I work as a fleet mechanic, who unfortunately has to deal with Ford. So far I haven't had to do much more than oil changes on the '21s that are starting to trickle into the fleet but they do something equally retarded.
When I pull anything on the lift I have to open the door and "sight it in" as I pull between the columns. Every damn time I open the door on a 21 f150 it puts itself in "park" right then and there. I can't hold the shifter in drive, the trucks in park before the shifter gets all the way to park position.
Anyway something to check out for your problem, something I see fairly often, (my mom's 98 intrepid did this too) is the park brake shoes within the rotor "hat" like to come unglued from the metal backing. Usually the friction falls off as I remove the rotor, and no harm done. But sometimes those linings wedge themselves in there and yeah it'll lock up the wheel. Switch directions and it dislodges, and you can roll.