it sounds like you have a wiring problem. unplug the dim headlight and verify your power and grounds. The right turn coming on may be a circuit problem or may be a wrong bulb. normally when you have your headlights on the parking lights come on. If you have a single element bulb in a socket for a dual element bulb you will bridge the two electrical contacts in the socket. Has some one recently changed a bulb?
Headlight should be on a completely separate circuit from the engine systems. However, electricity always looks to complete a circuit. Sometimes making connections through unimaginable ways. You may have to go and check each part of your headlight system to find which parts is affecting your engine. you may have to unplug your headlight switch and apply power to see what part will shut the engine down.