Wrap a bare copper wire around the back of the bulb socket if unpainted. Run that to the battery negative or to a good ground. Test with an ohmmeter inline to be certain before fastening it down. Then turn on the parking lights. The bulb should light if everything else is fine. Once it lights as a parking lamp, it will flash properly.

Check your front bulbs to be sure they are the correct type (two terminals). Temporarily replace with a known-good lamp (out of the taillight will be fine for testing). Then check your sockets to be sure both contact pins are properly springing back. Then insert the bulbs, try both ways, to gauge the quality of the sockets. I have seen some sockets so poorly constructed that the bulb will insert either way, and the wrong filament will light (the replacement sockets in my '69 tail lamp harness are this type...on my to-do list). Then be sure the sockets are securely soldered to the housings and don't wiggle.

I just ran into this issue on my dad's '68. It has the early two bulb front lamp housings, and they were heavily rusted and one bulb was shattered. Although they would flash on hazard, the right turn signal would not flash due to the bad bulb. And that bulb (and its companion) is really, really stuck in the socket.