This problem comes up pretty regular, and describing it is difficult with text, so I thought I would put together a quick solution to the problem of both turn signals blinking when one side or the other is turned on.
Here is a quick schematic of the turn/parking lamp circuit. Notice that each bulb has two filaments, and that both filaments basically ground to the same point. Also notice that the park lamps are fed by a common purple wire.
So the problem comes when one of the grounds is weak, or non-existent. This marked up schematic shows what happens. The current path is shown with the arrows. Basically a parasitic path to ground is found through the opposite bulb socket, passing through the parking lamp on the opposite side on the way.
What are some sources of weak ground? The obvious are a broken wire, bad attachment point to the body, or as in this picture, something less obvious.
There are no less than 3 mechanical interfaces where resistance builds up over time in the lamp housing. Try ohming out from the inner brass socket to the end of the ground wire. It should be almost 0. Mine was infinite. I fixed mine by taking my plumbers torch and solder and brazing the metal together at each interface.