Just an addendum to your question - these only really faded in the U.S. (and according to /u/jimmybabs are still in use here too), so it might be better to ask why they declined in the U.S.

Vertical parking lots are still built and used in Europe and Asia, and they are all over Japan.

First, these were never all that commonplace. They were custombuilt machinery installed in particular locations, and in the era before solid-state electronic controls and sealed motors, were often finicky and unreliable in extreme weather. As autos grew longer and wider in the 1950s, many models wouldn't fit into the steel cradles designed in the 1930s.

A second generation of larger mechanical storage systems found some buyers in the 1950s/1960s. A good example of one plays a minor plot role in the movie Atlantic City. Again, though, the reliability of these custom-built objects became an issue when compared to the simplicity of self-park decks created from standardized precast concrete components. And in most US cities, downtown retailing and office demand declined during the era—and lots of buildings were demolished for that and other reasons—so surface lots met a lot of the reduced demand.

A third generation of computer-operated mechanical parking operations can now be found in various places around the world. For some reason, there are quite a few serving the underground parking facilities along Prague's Náměstí Republiky. I haven't seen too many North American installations, but here's a video (with loud annoying music) showing animation of one under construction in West Hollywood. A search for automated parking system will bring up several potential vendors and images of existing installations.