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There is no way to have the running lights to come on anytime the key is turned on and the engine is running, besides of actually turning them on. There are several modules that control the lights, and also the BCM (Body Control Module), which is actually the dash fuse box. Each corner light has its own fuse and circuit for the BCM. Rewiring the circuits will cause trouble codes and module problems which can cause concerns with the lights down the road.
Just the headlights, not the parking lights. And the daytime running lights are just a module which pulses voltage to the high beam circuit to illuminate the headlights.
Please do not confuse this with the auto lamps which turns all the outside lights on when the sunload sensor on the dash senses the sun light going away.