Do you have twin element lights in the park light housing? If not then the park lights are just park lights when the light switch is in that position. They become turn signals only when the light switch is all on or all off. Two functions and only one light. When the light switch puts voltage to the park lights the flasher can not turn them off. Add a double contact base to the housing and move the turn signal wire to the brighter element keep the park light wire for the dimmer element and you will have both park and turn signals.

Steve H