67 Barracuda Turn Signal Dielemma

Ken. I hope you like puzzles! :)
This one shouldn't be too difficult.
It sounds like the parking light was a cross wire short to the turn signal.

There's an explanation of the turn signal, stop and hazzard circuits here:
1967 Chrysler Imperial Electrical Accessory Circuit From the Master Technicians Service Conference Session 236

A little explanation about the schematic they use.
The portions I've colored represent the moving brass contacts in the turn signal switch. All three move together when the signal lever is moved.
The black circles are connected to the wires and do not move.
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Might check whether one lamp is weak which would be another clue as to where a short gould be occuring.

Current flow with parking lights should be along the dashed paths.
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Something like this is going on.
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I'd probably check the bulb and the socket and wiring.
Do a search for posts about removing the parking lamps on a '67 Barracuda to get a sense of how its accessed.