I'm having the *exact* same problem. Originally I was having some issues with my car stereo. I bought it used but the CD player in it was not working, so I had a friend install an Alpine CDA-117 for me. It worked fine, but after a few days I noticed it started to shut off and turn back on whenever I turned on the A/C, pressed on the brakes, etc. I had the cables rewired and grounded to a second spot. This fixed it, but anytime I turned on the A/C the lights dimmed on my dashboard. I got frustrated with the problem so I took the car in for a load test and everything turned out fine although the battery was a little weak and got a new one. That same day, I noticed the problem youre having. Had to drive 60 miles without my dashboard or the rear lights on.
Did you by any chance figure out what the problem could be? I've heard that the Dash lights, Taillights, Parking lights are all on the same fuse (19--10 A, under the hood) and could have been blown. Was that it?
Thanks!!