Things can get very confusing if you make the wrong assumption....
The dual element bulbs have three connections: Two independant contacts at the bottom and a common connection on the "cylinder" part.
The way a typical car tail light is wired is that the common connection goes to ground and then + 12 volts is applied to the independant contacts to light the respective element.
An easier, cheaper way to accomplish the same task is to connect +12 to the common connection and then connect the ground to each independant contact.
If you, or some previous owner, rewired the tail light socket and made the wrong assumption then exactly what you describe would be the result.
Do you have a multi-meter? If so, run these tests:
Get so you can measure the voltage on all three wires. Leave the bulb out of the socket for right now.
With the headlight switch OFF (but capable of being turned on, so if that requires the "ignition" switch on or similar have that on) check for voltage (connect the negative meter lead to the NEG terminal of the battery) on all three wires. If there is no voltage then you probably have what I call the "conventional" bulb wiring.
If you see 12 volts on one wire then you probably have what I call "inverse" wiring. To confirm, switch the meter to OHMs and measure resistance to ground on the other two wires. With the headlight switch off and the brake switch off both should have a very high resistance to ground. Turn the headlight switch on and see if one wire goes to very low resistance. Then step on the brake and see if the other wire goes to low resistance. Once you have identified the proper wires make sure that the 12 volt wire connects to the outer "clylinder" of the bulb.
If in the first step you didn't find any voltage with both headlight and brake "OFF" then switch to OHMs and see if one (and only one) has a low resistance to ground. This would be a common ground connection and should connect to the "cylinder" part of the bulb.
Good luck!
Rod