As previously noted, Americans say “parking garage” for a building and “parking lot” for an on-the-ground parking facility, but the British “car park” is occasionally found here. (Commenters suggested that Grand Theft Auto, whose script apparently uses it, might have been an influence.)
The term popped up just today when I downloaded an app from a New Jersey real estate and parking company, Nexus Properties:
notoneoffbritishisms.com 
Could this be so as to avoid the use of the two separate US terms that you mention?
That would seem to be the logical answer, as space is limited in such announcements and there seems to be no term in American English that covers all sorts of car parking areas.
It will enable people to be sneaky in future. When people complain about the lack of parks in inner cities, reactionaries can say ‘actually, there are loads* of parks.’ *Do Americans say ‘loads’ ? How about ‘heaps’ ?
Yes to loads, no to heaps.
If we think of lots, or loads….. or heaps…… or whatever….
An ancient medical term for a distinct blood disorder has now become a cliché – at least on this side of the pond – I refer to ‘a plethora’, now meaning a surfeit.
But, if we are straying into the territory of collective nouns for too many car parks, how about this for starters:
An “ irritation” of car parks, or where AI card-scan payment techniques prevail, ” a denial” of car parks.
Hwyl Fawr!
Despite the creep of US terms in the UK that have increasingly caught on mainly as a result of their adoption in advertising (for example, ‘in-store’ instead of ‘in the shops’), the term ‘lot’, as ascribed to somewhere to park a car, remains off the board here.
In ordinary conversation, a ‘car park’ can be anywhere that’s designated – other than the side of the road – to leave your wheels.
Specifically, though an open-space ‘car park’ is precisely that, while one that is indoors over more than one floor is a ‘Multi-Storey’….. car park.
Here, a ‘garage’ is an enterprise that repairs said car if it goes wrong, and/or sells petrol…….
Alternatively, it’s a house-extension, somewhere to park your car overnight…. or store stuff until such time as it is converted into a spare bedroom!
My father always parked his car in the garage, even when he had to rent one that was two streets away from the house. I now live in a eight-house cul-de-sac with six lock-up garages and it appears nobody parks their car in the garages; they use it for storage and leave the car on the road. I often see eight cars outside, and I don’t even have a car. I’m told modern cars are too big to fit in garages built in the sixties.
Very often they are too big to fit in the lined bays in car parks.
It’s certainly the case that the 2000s Ford Focus (not a particularly large car) that I had up until a couple of years ago would fit into my 1960s garage but only if I folded in the door mirrors first, and then there was sufficient room for the driver’s-side door to open about 1/3 of the way and the passenger-side door not at all. I was glad to be a slim person at that time.
In 1963, we moved to a house that had a built-in garage. At that time, we had a Mk VII Jaguar, not a small car by standards of the time. My father could get that into the garage no problem, and get in and out.