No key in step on brakes, headlights go up and when parking lights are on head lights on or off the taillights have both elements lit up in.

For the second part, two filaments lit up is the classic sign of a bad ground. To verify that, if you have the older 1157 bulbs with the brass base, measure the voltage on the base. You should find 0 volts. If the ground is bad, you will find some voltage; most commonly around 6.0 volts.

I get the impression you have hidden head lights, and those open when the brake lights turn on. Most likely the relays are seeing the voltage on the taillight circuit and are operating in response.