As SAW said, globes are the first thing to check.
Swap it with a known working globe even if the filament looks to be intact. Make sure you are getting power to the bulb holder, and make sure the contacts aren't covered in rust and gunk.
Before getting too drastic, get your hand up behind the light assembly and just jiggle the wires around. Every so often my right rear indicator stops working, with everything else ok - a quick jiggle and it works again. As soon as I get time I intend to take the unit out and replace the crap wiring. It is pretty damned annoying.
They do have different power sources yes or else your indicators and park lights would come on when you brake.
I would try and clean the connections as stated above and swap your globes around as they can be blown but not black like you expect. Just hairline crack in the filament.
I would also be inspecting your trailer plug as sometimes they short out. Just open it up and have a look if anythings shorted or touching.
I have recently rewired my tail lights for this reason. Funky s*** was happening to mine.