Possible flasher?, I would take another look at all the bulbs that work in parking mode and in turn signal mode on the left side. After you see where each one is, then do the same on the right. I think you'll find you overlooked one or didn't check them while they were in the parking mode. A turnsignal/parking bulb has 2 filiments, one for each situation. Post you results.
It is definitely because a bulb on that side (left or right) is burnt out. I had a 1990 Cutlass, so I know.
check the bulbs on the side that does not flash, it's quite possible one is burnt out
One of the bulbs is burnt out. Replace the bulb.
Your "turn solenoid" is defective.
Check bulbs Replace turn signal flasher
One or two front turn signal bulbs is busted. Check your front turn signal bulbs and replace it.
If the turn signal bulbs work but just do not flash, replace the Flasher Relay.
One of the bulbs are broken, check all of your turn signals.
one or more bulbs for the turn signal have burned out....try replacing the bulbs
If the bulbs are good buy it just will not flash the flasher relay is defective. Replace it.
One of your turn signal bulbs is burned out. Either the front or the back.
Replace the signal flasher relay assuming you have physically seen front and rear, left and right signal bulbs flash.
Usually it's because one (or more) of the signal bulbs is burnt out. It is also possible that the turn signal relay is faulty.