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.
Try applying parking brake
Bad socket? Bad ground? Problem in steering column 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
if you recenly replaced bulbs u may have installed them backwards they should be running lights and brake lights if this is the case they maybe installed incorrectly
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.
You have blown a fuse, look for an acc. fuse out.
Possible ground issue at the taillights.
You can only turn off the headlights by applying the emergency brake. When the emergency brake is on, the headlights will not light. Otherwise the best of my expererience is that they are on all the time. If you have remote start, put the emergency brake on before you leave the vehicle and when you remote start it, the headlights will not be on, but the driving lights (if switch is on) will be on as well as the parking lights.
check the fuse, brake lights have their own, & check the brake switch at the top of the brake pedal
If you have no brake lights or tail lights then you've probably got an electrical problem. Check the fuses first and see if you have one or more blown.
I take it you have daytime running lights , if so by engaging the emergency / parking brake the daytime running lights should go off