I would guess that with the buzzer and bulb in parallel, what happens is that the bulb acts like a very low resistance (near zero ohms) so effectively it absorbs all the power and the buzzer gets none. If the bulb has a resistance of 1/2 ohm, and the buzzer is a maybe 600 ohms...the voltage divides proportionately and the buzzer won't sound.
The simplest most reliable solution will cost you about five bucks more. You disconnect the bulb, and replace it with an inexpensive 12-volt "dual pole" (DPDT or DPST, either will do) relay Wire up the relay coil to where the bulb was. The connect a 12-volt power lead to one side of both the relay's "NO" (normally open) contact pairs. Connect the blub to one of the paris, the buzzer to the other, and then connect them to ground. (If you buy a relay designed for car fog lights, there's usually a diagram included with it.)
The relay "goes on" instead of the originallight,and it isolates the light and buzzer so each one acts without interference from the other. That also will allow you to use a much more powerful buzzer.