Check the bulbs they are double filamented with smaller being for brake lights
Possible ground issue at the taillights.
they are not brake lights they are fog lights and they turn on because it is for the car behind to see you
Check the fuse for the headlights. It is probably open.
Fuse for parking lights probably blown.
Often this type of behavior is caused by a bad ground for the lighting circuit. Check the ground circuit for the brake lights.
ummm... your brake lights should ALWAYS work whether your headlights are on or not, that's how people behind you know you are stopping.
Brake lights should NOT be effected by the headlights. Brake light function is completely independent of the headlights. Assuming you are not refering to the tail lights, or marker lights, which are supposed to come when the head lights are turned on. Not having brake lights when the head lights are on is a big safety problem and should be corrected.
the brake lights are on a separate cuircut,if your brake lights are staying then it will be a fault with the brake light switch. the switch is located under dash,and will be on or very close to brake pedal.
head lights
Most likely a fuse.
The brake lights and the tail lights are separate circuits. Check the brake light fuse, then check the brake light bulbs, then check the brake light switch.
Is it the taillights or the brake lights? If taillights it may be a wiring problem, If brake lights check the switch on the brake pedal