By on all the time do you mean it's on even when the headlights/parking lights are off and your foot isn't on the brake?
The lights all chain off each other (driver to passenger) so it doesn't make sense that one would be lit and not the other, if the are wired correctly. I would check the contacts at the bottom of the sockets using an ohm meter. There should be 2 connectors at the bottom of each socket, one is for the headlights/parking the other is for brake. If wired correctly there should be almost no resistance between the same contact in the driver and passenger socket.