Parking light and blinker use different filaments. The bulb should be an 1157 with two filaments visible in the bulb envelope and have two bumps on the bottom of the bulb. Because the bulb has two different elements it has to be inserted correctly to get a bright flash and a less intense power saving parking light. The spades on the bulb are offset at different heights so that the bulb will only go in one way.
A blown dash light bulb will also cause the circuit to be broken and you will not get a flasher to work. If you have two bulbs in the rear (two brake lights per side as on the Impala-Caprice) one bulb could be blown and the system will still flash, but on a Biscayne-Bel Aire any single bulb failure (whether open from being burned out or a bad ground) will keep it from flashing.
Usually on these old cars water getting into the lights causes them to corrode to the point where the pig tail has to be replaced (light socket with the two wires leading into it).