Parking is used to describe a general location where cars can be parked (There is parking out front.) or as a verb (I’ll be there shortly; I am parking). A parking lot brings to mind a flat, open surface area. A parking garage is a building with multiple levels that holds a number of vehicles depending on the size of the parking garage.

You would not say the first one. Although it is is gerund acting as a noun, you would not use it to describe a parking place. It would be better to use 'Car Park'. In the other 2, the word is acting as an adjective, it is describing 'lot' and 'garage', making it a compound noun. These are more American usage than British English.

Thank you.

English, in America, would not use the phrase "car park." A parking lot will always be at ground level and is typically referred to as the place a person parks outside of a business, but not always. A parking garage, as noted, typically refers to a building with multiple levels, above and/or below ground and only used to park vehicles.