One wire is hot(power) all the time. That wire is most likely getting power from the lighting switch under the dash, but only as a common connection electrically. It could be connected anywhere that has power all the time. I do not know if it is controlled by the circuit breaker that is built into the headlight switch. It may be. Voltage regulator has nothing to do with power to the lights.
You can determine which wire is which by unhooking them one at a time, and testing for 6 volts from that individual wire to ground .
Once you determine which is which you could leave the one that goes to the lights in the back unhooked. Then before getting all froggy and running under the carpet, or under the vehicle, just run a test wire (16 gauge or heavier) outside the vehicle to the back eliminating the existing wire. There is also a disconnect (insulated bullet connector) in the trunk somewhere near the high brake light (follow the wire back from the socket to locate it) that can be unhooked to eliminate the existing wire from front to back completely for your test. I would unhook both ends.
Personally I do not think that there is a problem with the front to back wire. You are using that same wire when you say that the lights get brighter when you unhook one light. Your problem is in the back with the bulbs,or bulb sockets, or modifications, or connections. Clean the socket spring clips down to bare metal for good grounds. Make sure you have not wired up so that the front parking lights come on with the brakes.