I’m driving a 2010 Hyundai Sonata and just recently one of my brake lights has been acting up. Everything works perfectly fine as long as the head lights are off; however with the head lights on, when I depress the brake pedal my driver’s side tail/brake light goes out. The center brake light and passenger side brake light work just fine regardless of the head lights being on or off. With the headlights off, all the brake lights operate normally when the pedal is depressed.
Maybe a bad ground at the driver’s side light connection. With the headlights off, the brake lights can ground through the tail light circuit and to another light that is grounded on that circuit. But, with the headlights on and the tail light circuit is powered, it negates that ground circuit and nothing lights.
Check your brake light bulbs, years ago I had a dual filament bulb go out and the broken filament ended up attaching itself to the good filament. When I used the turn signal, power went backwards through the attached filament with strange lights working and caused me all sorts of headaches trying to figure it out.
I agree with @BustedKnuckles advice. The grounding for the rear light having the trouble needs to be looked at. The headlights use much more current than the brake light does so rear light doesn’t get the proper current it needs to work because of the voltage drop across the ground system when the headlights are on. When the Hdlights are off there is no voltage drop to rob the power needed by rear light. Check and clean all the grounds under the hood.