All my rear lights don't work, including; running, brake, turn signal, and license plate stopped working. I do have an original dealer installed towing package. All lamps were inspected and appear good. I have tested interior fuses as well as applicable fuses in breaker box inside engine compartment. I don't know how to determine if fuse relays #37, #38, and #39 are good. They seem to be warm to the touch. I do not see anything visually impressive when checking the wiring assembly in the rear of the truck.

 

I am going to take a SWAG that the ground is the problem, but I am working from a 2009 EVTM. This should be the same, but it might not be.

The lights you list have a common ground lug, common splice and common connector.





You could test this, if the trailer tow adapter is in good repair, as the ground on the trailer tow adapter is a different ground, in the 2009 EVTM.
Test from the + VDC pin on the license plate to the ground on the trailer tow adapter to see if you get +VDC when the parking lamps are on.

Alternately you could get the resistance reading from the trailer tow adapter ground to the ground pin on the license plate bulb socket to see if it is < 100 ohms.
- Parking lamp bulb socket is also valid for these tests. Dealer's choice on which is easier to test with.

This information is from the 2009 EVTM, so it might not be 1:1 the same as the 2010, but they are in the same production run ( like a 1997-1998, 2001-2003, 2004-2005, etc ) so they should be similar.
- Ford could do some strange things with production runs that should be the same, but have 1 or 2 odd ball things are are nothing alike...

Worth a check, without any specific 2010 suggestions from 2010 owners that saw the problem 1st hand.
- I have having a day with replies only being SWAGs, I feel useless for answers...