Check all bulbs and wiring . I sounds like you have the brake light system crossing into the parking lights.
Something I recall from the past and worked on all my dads cars pre 70. He would aways take the Keys and leave us wait in the car. I would always get behind the wheel and pretend to drive . I figured this out, and it would kill his battery. If you would put the 4 ways on they would work without the key so they had battery power all the time. The turn signals would not so they were tied into the accessary on the ignition. So I put the 4 ways and the turn signals on the one day in the rain and the the wipers started to move and stop with the blinkers. so I tried other things . The heater fan radio and anything tied to the accessary would go on and off with the blinkers. Then I hit the brakes and low and behold everything in the car worked and stayed on steady as long as you would stay on the brake have the 4way and blinker on. Well that was it, every time he left us in the car we would listen to music and activate different accessaries. "We were cruisin" My dad would always have to get a jump start when he would come out. He bought a new battery and we killed it. We never told him. This was driving him crazy for a couple of years through three cars that I remember. A 65 426 4pd conv fury. A 66 crysler 300 conv. and a 69 4sp corronet R/T. This worked on every car he left us sit in while he was doing what ever. Me and my 4 sisters kept our secret. He went to his grave not knowing. So this is what a back feed of power could do. Not saying this your problem. But just a freindly FYI and a memory from the past