There are fuses for the running lights, but they're those big cartridge types, so I don;t know if there are fuse taps available. This might help.
New 5th-Gen 1500 Fuse / Relay / Circuit Breaker Diagrams & Documentation
Update: I was intrigued, so I poked around a little. Underhood fuse f12 is labelled for the trailer running lights. I pulled it and checked voltage. One side gets power with head or park lamps on. I'm guessing the other side goes back to the trailer connectors. I don't know if anyone makes fuse taps for that style of fuse, but it looks like someone with a few male and female spade terminals, some stiff wire, and a soldering iron could easily build some extensions to elevate the fuse a little and tap onto that.