Brakelight don't work when car running

You simply have a burned out bulb or bulbs. With the key turned on turn on the directional for the left side and then the right side. Typically one bulb is blown and the opposite side works normally. Observe the normal side and note which is the flashing bulb. It is the flashing bulb on the other (malfunctioning) side that needs to be replaced. To replace the bulb you must remove the entire brakelight assembly, which isn't difficult. Open the trunk and locate the electrical harness that lines up with the brakelight assembly. Near the harness, locate the two (possibly three) nuts or wingnuts that attach the brakelight assembly to the car. Once the nuts are removed, the assembly can be pulled off the rear of the car. After replacing the bulb, be sure to test the operation of the new bulb before reassembling everthing.

Check the bulbs.

Take the lens off of the light and change the bulb, it is probably burnt out.

Check the fuse in the fuse panel under the dash. Check the brakelight bulbs and see if they are good, it is unlikely the bulbs will both blow at the same time, it pays to check them. If the fuse is good, then the brakelight switch is bad under the dash,on the brake pedal. If you haven't done these repairs before, it is very tight getting under the dash to get to the brakelight switch. You may want to take it to a shop and let them do it.
I hope this has been helpful.

Auto parts stores and others should have that standard sized bulb.

the bulb will need to be replaced. It would be best if both are about the same age. If bulb is new, then the brake switch is faulty.

Some makes and models have different taillight bulbs and brakelight bulbs. in other words your turn signals and brakelights are on one bulb and the tailights are on another, or they have multiple filaments and one of them could be bad

normally there 2 screws on the lense, usually going through into the body. phillips head or torx head. just take those out and pull the unit straght out, and the bulb is right there. if you have a cargo light on it, those bulbs will usually be on the outboard side of the brakelight, on either side of it. once you change the buld, wipe the new one with rubbing alcohol to remove the oil from your skin. the bulb will last much longer if you do that

You have a bad ground conection.Remove the plug from the lamp unit.Check and clean with WD 40.Make sure that the bulb holders have not rusted away.If so replace.

Test for power at the inoperative bulb socket with a 12V test light