There is a sensor on the transmission at the shift arm called the MLPS (though some auto parts stores will call it a transmission range sensor or neutral safety switch). That sensor is attached to the transmission case via two small 8mm or 10mm head bolts. Either the sensor isn't aligned correctly (the bolt holes are slotted), it has failed, or the wiring to/from it involving the reverse lights is faulty. I'd start with confirming that it's aligned, then either break out the multimeter to test for power, ground, and continuity at that plug (you'll need a diagram), or replace the sensor.