My 1995 Ford E-150's running

I think it would be the wiring near the rear. If it has trailer wiring then check all those connections. Not posible for a swith as on your vehicle the running lamps is one wire until it gets to the back. Also be sure and actually check for voltage at the bulb connection.

Take out the taillite and check the condition of the wires. especially the ground wire thats screwed to the sheet metal. a little corrosion can cause this. Also try wiggling the bulb in the socket with the running lights on and see if it starts lighting. Ground wire will usually cause other strange symptoms in the taillights, so I'd bet the problem is confined to the bulb or its socket. you may need to change the socket, Inspect it closely for dirt/corrosion. Don't forget to post your results. Cheers.