Could be a couple of problems, one lose wire in pig tail harness, but since you say its both back up lights it may be a relay switch when you place into reverse the switch does not respond to activate the lights
Try checking the neutral safety switch. This switch is the one in automatic transmissions that makes the backup lights "flip" on when going from park to drive as you go through the gears. When my neutral safety switch shorted my backup lights stayed on constantly.
Check the backup light switch.
Start with checking the fuses and bulbs. See sources and related links below for bulb information.
you should try checking the fuses and replacing the bulbs the fuses for thos lights should only be about a 10 or a 15
Start by checking your fuses. Then bulbs. Then switches in that order.
Check Fuses, Replace Bulbs, Replace Shift Position Sensor. End of solution
It could be caused by a bad ground or severed wire somewhere.
check your bulbs and fuses check your bulbs and fuses
Start with checking the bulbs, then the fuses. In most vehicles the brake lights and siganal lights and running lights are separate circuits. If the fuses and bulbs check okay check the wiring feeding the rear of vehicle.
Check the fuses Check the backup switch Check the wiring - make sure that there is good contact.
have you checked the bulbs and fuses
use a test light and check for power at the bulbs. if you have power then you want to be checking for a bad ground. If no power trace the wires and look for shorts/breaks. hope this helps!