Don't know about your truck, but in our Dodge, there is a relay in the tail light circuit that provides power to the trailer running and tail lights. The power comes through a different fuse at (or near) the main panel. The theory behind this is that a lot of running lights, etc. on a trailer would pull too much current through the truck's headlight switch and burn out the switch.
In any case, when the parking lights come on in the truck, power from that circuit closes the relay. That provides power to the trailer plug for its running and tail light circuit.
Therefore, with one bad fuse or one inoperative relay, there will be tail and parking lights on the truck, but nothing at all on the trailer.
Suggest you check your truck's trailer connector to see if there's power to the correct pin when the parking lights are on.
These people have great wiring diagrams and drawings of which pin in the connector goes to what.
Trailer Wiring Diagrams | etrailer.com