But if the headlights come on by themselves, then that's not the problem. The parking brake triggers the headlights off, which keeps the dash lights off as well. When you manually turn on the headlights, the dash lights come on with them.
I had a guy come in yesterday with the same problem in a Regal GS. It ended up being his dimmer switch had gone out. The radio and climate controls would black out, along with the dash lights not coming on, when the headlights came on, and adjusting the dimmer did nothing. He replaced the switch, and all was well.