Model Cars

They weren't props: all the cars (real, experimental and model) used in the film were the real thing, being developed by the Ford Motor Company, which gets a thank-you at the end of the film. What's interesting is to see how none of these innovations ever made it onto the market -- including the vehicles in the lobby showroom at the start of the picture (cars of a future that never was) as well as the parking model you mentioned.

I'd like to think these cars and models are saved in the Ford museum or some similar place, preserved in some fashion. But it wouldn't surprise me if a lot of these things were simply destroyed or discarded, as was often the case back then.

The models themselves, and the non-operating mockups, often were destroyed later on by the design studios when they needed the room and the design had sparked no interest among the corporate guys. Many of the operating prototypes or "car show queens" were retained, making the car show circuits (sometmes redesigned). Some of those survived or went into private hands, fortunately.

The black convertible (sort of) that they are driving around New York is a Ford concept car, usually called the 1953 Ford X-100 -- sometimes known as the Lincoln Continental 1950-X or the Lincoln Typhoon 1957, probably with some minor design modifications along the way. It's now in the Henry Ford Museum in Dearborn, Michigan.

You can easily see that some of the X-100's "torpedo" styling did get used later on, such as in the 1961 Thunderbirds and Ford Galaxy series.

Locating and researching old concept cars is a small but growing sub-genre of antique car collecting, and they do turn up now and then -- often in dreadful condition, rusting away in some barn or junkyard.