The quick flashing is consistent with a bad/missing bulb. If the other side flashes normally, this is not a flasher unit problem.
Do you have a test light? If not, you can likely purchase one for about $10 at Sears or Radio Shack. Clip it to a good ground. Then probe the terminals inside the socket (being careful not to touch the outside). One should light the lamp when the parking lamps are on and the other should flash when the blinkers are on. If it flashes, you are looking at a ground problem (likely in the socket itself) or poor contact between the socket contact and the bulb. If the lamp does not flash, you have a wiring problem on the power side of the circuit to the socket.