Q: Left front blinker won't blink when the lights are on or parking lights

Be sure the bulb filament is intact, although if the turn signal works, as you say, when the headlights or parking lights are off, the filament is probably OK. For some reason, the turn signal circuit is being interrupted by those other two circuits. The only way to figure out where and why is to trace the turn signal circuit using the factory wiring diagram. If you trace the circuit when commanded "on", using a wire trace you will definitely find the interruption. If you desire that the required diagnostic steps be performed by a certified mechanic, please request an electrical circuit/component diagnostic and the responding certified mechanic will get this resolved for you. If you have further questions or concerns, do not hesitate to re-contact YourMechanic as we are always here to help you.