Could be the GEM module, or maybe battery low voltage. The GEM tends to do weird stuff when the voltage gets low, although I've never heard of lights coming on. Wipers, horn, etc, but not lights.
Maybe there's a short between circuits that heats up the light sensor that's normally dead when the key is off.
Maybe the truck's just afraid of the dark.

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First truck -- 1929 Model A Ford pickup, restored from ground up. Wish I still had it!
99.5 F250 PSD Supercab LB 4x4, 6spd, 3.73LS, Boost & pyro gauges, Swamps S175/146 injectors, Jody's 80 HP Econo PCM (classic version

), AIS,
coolant filter w/"hokum" bracket, regulated return,
heated mirror mod, lighted cupholder, Marinco heater plug-in.
Hard or no-start? Check here