You can get some really strange electrical problems if a dual-element bulb fails in a certain way. The parking light / turn signal bulb on the front sides is a dual-element bulb.
The failure mode is when the filament that is higher burns through, and the remainder then falls onto the lower element. It can then become stuck there and you get a short from the parking light element to the turn signal element. This causes the strange interaction.
Remove the parking light / turn signal bulb from each side and examine it closely. This bulb should have two distinct elements in it that are not touching in any way.
Judging from your turn signal indication when you turn on the parking lights, its the bulb on the right side.
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Ken300D