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.
It could be caused by a bad ground or severed wire somewhere.
Check Fuses, Replace Bulbs, Replace Shift Position Sensor. End of solution
Check all the fuses. If not bulbs or switch that's all that it can be,
check your bulbs and fuses check your bulbs and fuses
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
Likely to be the brake light switch on the pedal at fault if the bulbs and fuses are good and other lights work ok.
you could have possibly bought the wrong bulbs and/or fuses. or when you were replacing them you may have damaged the wires. Also the reverse light switch may be defective.
Check the fuses first then check the bulbs.
If all the fuses and bulbs are new and work properly, the lights could be malfunctioning due to a short in the wiring. The wiring may also be damaged or burned out in places.
Check the fuses Check the backup switch Check the wiring - make sure that there is good contact.
the bulbs are probably burnt out, though it could be a bad connection or sensor or fuse or sveral other things. check the bulbs and fuse first though check your bulbs, fuses, and connections,