I have a '95 dually and it has small yellow running lights on the rear fenders. The only time those lights will come on is when the headlights are on (or parking lights) and I put it in reverse. Not sure, but I would have guessed they are supposed to on when the lights are on and go out when the truck is put in reverse.
All the ones I have seen are on all the time the headlights or park lights are on. They should not go off at any time unless you turn the other lights off.
I suspect a wiring problem at a trailer plug. Seeing this is a dually, I am sure you probably have trailer plugs, and may even have one in the bed for a goose neck or 5th wheel. Some people's wiring practices are not very good if these where added in later.
If the ground for these little lights happens to be hooked to the reverse lights, they would go out when the reverse lights are activated.
Thanks for the reply. I'll check the things you mentioned. My assumption has always been that these lights would/should go out when you put the truck in reverse because I am blinded by them while backing if I'm using the mirrors, which is just about always. If that's not the case, I may try to figure out a way to set them up that way.
Anyway, thanks again and I'll let you know what I find out.