The running lights are fed separately from the brake/turn. The first thing I would do is make sure you have power at the connector by using a 12V tester. If you do, then you know it is a trailer issue. With all the parking lamps not working, it is more likely that you have a bad connection where the wire ties to the connector either on the truck or the trailer end. Start with the tester to eliminate or target the truck end, if there is power there when the truck parking lights are on, then test the wires coming out of the back of the trailer connector. If it is dead there, then you know it's in the trailer connector.
Daniel & Donna Leonard
2011 Ram 4500 SLT 4x4