Weird lighting issue

Hi folks!!!
My son has just bought a used 2006 Aveo LS with 133,000 miles on it. It is a "plain Jane". No cruise, power windows or power door locks. The car almost looks new, so someone took very good care of it.
Now to the problem;
With the engine off and the headlight switch full on, the headlights are not lit, but the left tail light and left front parking light and all the dash lights are lit but the right tail light and right front parking light are not lit. Start the engine and the headlights light up along with right tail light and the right front parking light. Turn the main light switch to the off position (engine running) and it goes into daytime running light mode but, the left tail light, left front parking light and the dash lights are lit. The right tail light, right front parking light and dash lights are now lit.

Confused??? I am!!!
Oh!!! If it makes a difference it came to us with an after market radio.
HELP!!! Please,

Walter

As I mentioned in my original post the headlights (not the daytime running lights) will only come on if the ignition switch is in the run position. Is THIS normal???

As soon as I turn my key to off to cut the ignitionmy headlights turn of, but my dash lights do stay on