What's a radio?
Seriously, sounds like you grounded the radio to the positive headlight wire behind the dash. It is getting it's ground through the light filaments, but when you turn on the light switch, the radio "sees" 2 positive inputs. The headlights take around 10 amps to light, so the radio uses considerably less juice.
Of course, this may have happened because a common ground is loose somewhere. For example, the screw where the radio is grounded may be the same screw grounding the lights. That screw is not making good contact with the chasis, but the wires are making good contact with each other. So, the lights are grounding through the radio. ..kinda the opposite of the obvious?!?
Still, my first step would be to re-ground the radio to something other than what it is currently grounded to.