Car park is the British English term, the North American English term is parking lot. A muliple-storied building where cars may be parked is referred to as a multi-storey car park in British English, and a parking garage in American English.
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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 lot (American English) or car park (British English), also known as a car lot, is a cleared area intended for parking vehicles. The term usually refers to an area dedicated only for parking, with a durable or semi-durable surface.
A multistorey car park (British and Singapore English) or parking garage (American English), also called a multistory, parking building, parking structure, parkade (mainly Canadian), parking ramp, parking deck or indoor parking, is a building designed for car, motorcycle and bicycle parking and where there are a number ...
A parking lot (American English) or car park (British English), also known as a car lot, is a cleared area intended for parking vehicles. The term usually refers to an area dedicated only for parking, with a durable or semi-durable surface.
In general: in is the most common way to say this, and you can use it really any time you are inside of some area with well-defined boundaries (e.g. in the parking lot, in the yard, etc) at is also fine, and would not sound particularly strange in most situations. on would sound strange to most people.
Australians say “garage” for the place where you get your car serviced, and for what you park your car in at your home. But places where you pay to park your car are called carparks, not parking garages.