There are two filaments inside a turn signal bulb. One of them is dim for the park lights, the other is bright for the turn signal. The bright filament can break on one end and land on the dim filament. Anytime the parking lights are on it makes both filaments light. The turn signal can still function depending on where the break in the filament is. This would also change the resistance in the bulb and cause the turn signal to flash faster.
As seen in the picture below, if the top filament was to break loose from one end it could land across the other filament causing the problem you describe.
First thing to check is the bulb.