I believe you are "standing", not parked.
If your car is running when parked then it is burning gas. Cars use gasoline to run the engine.
Exhaust fumes may enter and that is not good.
Generally speaking, the driver in motion is considered to be at fault when a parked car is hit, even if the car was parked illegally. The exception to this would be if the car was parked in a dangerous, hard-to-see location and the driver in motion could not have reasonably avoided hitting it. For example, if a car is parked on a sharp curve in the travel lane of a narrow road, and the driver of a car driving around the curve at the speed limit would not be able to see the parked car until too close to it to avoid a collision, then the driver might not be liable for damage to the parked car, and indeed the driver who parked the car might be liable for damage to the car that hit it.
driver of parked vehicle
If you are parked with the motor running, engage the parking/emergency brake. Turn the ignition off and restart the engine. The daytime running lights should not be on.
It's difficult to imagine circumstances in which the owner of the parked car could be considered to be at fault. Almost always, the driver of the moving car will be found liable. Even if the car was not parked properly, the driver of the other car generally will have had the "last clear chance" to avoid hitting it.
You don't unless you are parked with the engine running. If you are parked and you want the engine running but don't want the daytime running lights on just shut the car off and set the parking brake then start the car back up. The lights should remain off until you release the parking brake.
Diesels are kept running while parked for various reasons. In cold weather, a diesel may be very difficult to start so it is better to leave it running. Maybe the engine is needed to run heat or A/C for the sleeper cab.
If the other vehicle was parked, there was no other driver to have license, insurance or registration. The driver who hit the parked vehicle is at fault and is liable for all damages to the parked vehicle.
If the car was legally parked, Yes.
The driver that hit the parked vehicle would be at fault.
No it won't hurt the engine BUT you could get carbon monoxide in the car. Don't sit in a parked car with the engine running if your exhaust is bad - it could kill you.