Thanks so much to everyone for your suggestions. Ya'll are such a great pool of knowledge. Many things were brought up that I had not thought about.
Here, briefly, are the troubleshooting tests I conducted.
I've tested and re-tested my trailer's braking activation problem by ensuring there was no obvious shorts due to one or more strands of different wires accidentially "touching".
I tested several different wiring circuit fault possibilities by disconnecting different wires, then by driving on adjacent slow speed roads, towing my Airstream, while turning on and off the headlights of my TV. (I'm surprised deputies have not been called.)
The fault would only occurr when I turned on my parking light. Headlight activation was secondary to parking light activation and had no part in trailer brake activation.
The trailer brake activation only occurred when the black ground wire was very, very loose.
My trailer light lead cable is 'just" long enough to reach the TV connector.
That is not good and could be a source of the problem, but the trailer brake activation occurred during straight level towing, not during turns, tight turns or going up or down hills. hummm
I then stripped back the outer covering of my trailer lead wiring cable. I discovered that several of the seven interior wire insulators had multiple cracks/breaks in their insulator covering.
This may or may not have be the cause of the trailer braking activation when the parking lights were turned on, but it sure showed me that the trailer lead wire wiring harness was aged and faulty and needs to be replaced.
With the trailer wiring harness' black ground wire firmly in place in the 7 plug socket, I could not make the trailer brakes erantly activate under any condition.
I tried to research how a 1968 Airstream
24' Tradewind's lead wiring harness is connected, but I could not find an exploded diagram showing the path / terminal / layout.
At this point I am thinking about hiring a professional to re-wire the trailer lead wiring harness.