Could be a bad wiring hook up. How did you install the new radio? Did you use a wiring harness kit?
You could have tapped into the power wires for the dash lights, instead of the dedicated radio power source.
Or it could be just normal. I have an acura and thats how it is, during daytime when not using headlights, the dash lights are bright, at night time when I turn on headlights, the dash lights dim, because its easier to the eyes. This has nothing to do with the radio though.
Your dash lights should have a dimmer switch, set it all the way to the max until it clicks, dash lights should remain the same, with or without headlights on.