I do not know about 1937 Chevy's but my 1935 Buick had a tail light with a dual filament bulb socket with a dual filament bulb.
Pull the light switch half way front parking lights and tail light would come on.
Pull the light switch all the way headlights on, front park lights off, tail light still on.
If your tail lights are going off then they are being fed off the same terminal as the parking lights. This is why your tail lights are going off when you put your headlamps on.
Something is wired wrong at the switch, tail lights are fed from the parking light circuit which is wrong.
Last edited by 32vld; 07/20/2015 4:16 AM.
