Parking lot would be the US equivalent. While he was driving out of the car park (parking lot) ... or While he was driving away from the car park (parking lot) ... If it's a building in the US, it would likely be called a Parking Garage or Parking Structure. Perhaps Car Park is also used in the US but I think of it as BE.
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The driver will wait for us in/at the parking lot. All I know is that in the US, the 99.9999% assumption would be that the driver is waiting in the vehicle. If he's not, then that would normally need a special explanation. And there isn't one here. So if the vehicle is in the parking lot, he is in the parking lot, by extension.
After all, we drive into the parking lot. The parking lot is also a two-dimensional area, but it can be three-dimensional if the parking lot is enclosed (with a roof), which adds the notion of "height." In any event, I don't find "parked on the parking lot" incorrect.