Turn signal issue...

Bad ground. one of the turn signal lamps.. When your headlights are off the parking lights are also off. The turn signal grounds via the other half of the lamp.. Head and parking lights on that path is not open to them.

You look specifically at where the lamp wire pigtail connects to the trailer frame

OR

If a butt connector is used with an extension, at the butt connector.

Put a new wire end on where the ground wire connects to the frame. Then shine the frame with sandpaper until it is bright. I use KEPS NUTS with a screw. They are harder to come loose. Tighten the snot out of it. Then smear Vaseline over the nut and bolt both sides.

Still dim with a problem?

Replace the butt connector

Still dim with a problem?

The issue is within the lamp's socket

Some tail/turn light assemblies have 3 wires; turn, tail, and ground. Some only have 2 wires, tail and turn, and use the metal frame assy as ground. And the metal frame of the light assembly is bolted to some part of the metal frame of the trailer since the trailer frame is ground back to the battery.

If you have three wires, then determine which wire is the ground wire and trace it to where it attaches to the trailer frame. If only two wires, then make sure the light assy frame is clean and well connected to the trailer frame.

I would also check the ground wire coming from your cord plug to make sure that wire has a good connection to the trailer frame.

Thanks, Pauljday!
As a matter of fact, I had the tail light repaired in September and the job ticket says "found loose ground at the tail light assy, tightened loose ground and tested".
Obviously, the problem is back. What do I look for in the wiring?

Bad ground

YOu likely have a bad ground/negative connection at or near the failing bulb.......including a corroded socket.