Check if anyone felt like playing around with the wiring harness by the taillights. Seems like the only likely cause, especially if you have isolated the issue to just the reverse mode.

Are you the original owner of the car? If not, it's not unlikely that the PO didn't understand the bulb-out indicator, and may have done some creative wiring.

__________________
2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car