Electrical issues.

vpmopar said
Turning the headlights on also turns on parking lights. Turns on all taillights as well. I have disconnected the connector going to all rear lights with no affect on the issue. Ammeter gauge shows discharge when lights are turned on. Have replaced headlamp bulbs with no change.

OK good. I'll say a number of people here have found the front marker lamps on some pre-68 cars turn off when the headlights turn on.
This is simply the terminal the front lamp wires are connected to the headlight switch.

The headlight bulbs wouldn't be the cause of crossing with the turn signals.
The high beam indicator maybe a clue. We'll get to that. I don't show that schematic but we may have to add it.
Signal circuits are a little more complicated.

next situation.
Turning the key on connects power to the run and accessory feeds. Run is ignition and alternator. Accessory is everything else.

vpmopar said
Turn the key on and the turn signals work perfectly, right and left front and rear and the indicator lights on the dash work correctly as well.

Finally, lets look at the turn signal and indicator wiring and see if we can figure out what would work normally with the headlights off but get a backfeed with the headlight on.

I don't have a '64 shop manual, so if you do, that will help.