Yes the reverse sensor in the transmission and or wire to the sensor has issues.
What transmission does your 91 Explorer have?
This wire goes from taillights, up frame, near brake booster and through the transmission wiring harness. This harness goes through the upper intake and down the spine of the transmission (either 5 speed or auto) It is a single wire, black with pink I believe (would have to check)
If the switch in the trans is bad it will cause this. However to test the wiring first you can ground out the signal wire (at the sensor in the trans) and see if the lights come on.
Also I found some of these sensors are cheaper at the mazda dealer then the Ford dealer, some are a dealer item or used item only.
91 mazda Navajo or B4000 will have the same trans sensor.
Your truck should either have the A4LD automatic or the Mazda 5 speed.