Check the owners manual for the tow vehicle. Many have a seperate fuse / relay for operating tail / turn lights and for operating running lights. Very likely you have a blown fuse, in the tow vehicle.
I've had that running light issue, twice in the past. Once, towing with a Nissan Titan, and once towing with F250. Both times, it was a fuse / relay in the fuse box under the hood of the truck.
18 Nissan Titan XD
12 Flagstaff 831FKBSS
Wife and I
Retired Navy Master Chief (retired since 1995)