On my 1971 Camaro, the turn signal indicators are lit when the headlights are turned on. Furthermore, when the blinker is turned on, the indicator light behaves as expected, blinking in time to the turn signal. However, the turn signal and side marker lights are out of sync, first parking light blinks, then side marker (synced up with headlights off). In addition, on the rear, you can tell there is a feedback from the turn signal going to the other turn signal that is not engaged. Everything works fine and as expected when the headlights are OFF. All bulbs are new. New flex circuit board and refurbished gauges in the instrument cluster. New headlight switch, turn signal flasher, hazard flasher, and dimmer switch installed. Checked and cleaned grounds for parking lights and brake lights. Checked with ohm meter, all good. One other symptom is when the headlights and blinker are on, the tach (brand new) behaves erratically. It settles down when the blinker is turned off. This only happens if the headlights are on. Also tried with fog lights only and the problem is identical.
The bulk of the problem will be due to poor grounds to the front bulbs. Most of the time will be faulty front turn/park light sockets. You van easily test this with a piece of wire. strip both ends of test wire and touch one end to the metal bulb base while it is still installed in the socket. Just below the glass potion of the bulb. Touch the other end to a know good chassis ground. If you do this with the turn signals on you will notice that the flashing element in the bulb will switch to the other element and get much brighter. If you do this with the headlight/park lights on you will notice the interior turn signal light will go out.
If doing this test corrects the condition you will need to follow up with touching the test wire to the ground wire before it enters the socket. This may require stripping some insulation back from the ground wire. If touching the ground wire with the test wire makes no difference then you need to replace the socket assemblies. If it makes the same correction to the bulb operation you will need to trace the ground wires back to a poor connection to the body. Usually at the inner fenders or radiator support.