Which bulb?
 

Which bulb is giving you a problem? The one in the tail light assembly (the one that also contains the reverse light) or the one in the bumper if you have an SUT.

The one in the bumper is a 2 part piece. A socket assembly gm part 12335899 usually has a leakage issue where the bulb contact fail, another failure point is the 3 pin connector that actually plugs into the assembly where the weather pack connector fails and corrodes fails to provide power to the bulb socket. When ever working with bulb sockets or connectors be sure to use dielectric grease!

As other have said no power but you have 2 power sources and 1 ground, No parking lamp or no brake light (or turn signal) means no ground, plain and sure. Unless its a snow belt Hummer is highly unlike both parking and brake/turn would be corroded but possible.

Either way its no big deal. H2's have an issue with some chinese leds that can cause strange issues like cruise control not working, crazy turn signal flashing and other problems. Start with a known good working system then add led's from there making sure to check everything before going to the next target bulb. Happy Hunting!

license plate light .....check it.. wires shorted?? had exact same issue on my 2007, thats what it was...