So, we'll make the assumption your driving a Trailblazer/Envoy type of truck.
The plate lights share a fuse with the left rear tail light, fuse # 14 in rear fuse panel.
If the fuse is good (and the LR tail works), and the lamps are good, you probably have a broken wire in the gate wiring harness.
You'll need a meter. Turn on the park lights and check for 12v at the plate sockets. No 12v, the wire colour is brown.
If you have 12v, test the ground. No ground, the wire colour is black.
What condition is the inside of the sockets, if dirty and corroded, you probably need new sockets.