I managed to get a hold of a 1990 YJ Wrangler today, and so far the most annoying thing that I've found is that the aftermarket Kenwood CD player will lose all power when either the headlamps or parking lamps are turned on.
No fuses are blon on either circuit, and the Kenwood deck does not lose any presets.
The Kenwood looses power for about 2 seconds immediately after the lights are turned on.

Could it be possible that the last owner somehow managed to short a circuit, or connected the radio to the wrong circuit entirely?
Also, any ideas/advice on what troubleshooting steps I should take would be greatly appreciated.

I'm guessing here...There are three 12 volt inputs to the radio:

1. 12 volt supply to the AMPs.
2. 12 volt supply to the memory.
3. 12 volt supply for a powered antenna.

Now, the first question:

After the lights are turned on and everything returns to "normal" and you're toolin' down the highway... watch the display on the CD control head. If it's LED/LCD does it "Pulse" with the music... or does it gain or loose brightness as the low booming base notes sound?

Now a CD changer doesn't take a great deal of power.. but it IS sensitive to power fluctutations. When ever you pull on the lights the power does flucutate, and a short lowering of the source occurs as the regulator catches up. During this short pulse of lowered voltage the CD, though not brain dead (the memory still works) becomes unable to operate, spin or change CD's, etc..

I'd change the CD's power input (number 1 above) to another source. Were it mine, I'd install an auxilary fuse box off the batter and mount it in the engine compartment. I would wire the input to the fuse box directly from the battery and take the source voltage off one of the fused outputs.

The display on the deck does not dim in anyway, and since there is no external amp hooked up (yet). The system won't reproduce enough bass to draw enough amperage to cause the system to sag. There is no CD Changer is involved.
For right now it's just the Kenwood deck itself that the previous owner had installed.
I will try changing the power source of the Kenwood before I switch my whole system over from my Escort (Sony headunit, 400 watt amp, 2 10" subs...the usual) once I have this figured out. Though talking to some car audio guys, they seem to think that it's a problem with the Kenwood deck and the way that it reacts in the Jeep, so I may just swap the Sony headunit over and see what happens.

The secondary fuse box is also a good option as well. Plus it would allow me to hook up my CB and other knick-knacks from my Escort.