For the most part, of the approximately 60000 vintage American cars on the island, most of them are kept running though a mixture of ingenuity and mechanical improvisation. The American embargo meant that Cuban mechanics were cut off from US parts which forced Cuban mechanics to extemporize. Cannibalizing parts from broken-down or wrecked cars is still a major grey market industry and a number of foreign observers have noted that Cuba lacks junkyards. Occasionally, Cubans were able to acquire American parts from third-party markets like Latin America, but this sometimes meant having to adapt parts from different manufacturers.
Agreements with the Soviets during the Cold War led to exchanges of Soviet or Eastern bloc automobiles like the Lada or Skoda. Cuban drivers for the most part found these automobiles lacking compared to American cars, but they were a source of fresh parts for the aging fleet of American-built models. Cuban mechanics have been proven adept at mating Eastern-bloc engine components to the old school Detroit cars. For example, some vintage cars use Lada carburetors or a Volga diesel engine.
The result is that the appearance of so many vintage Detroit cars functional might impart to Cuba an appearance as an automobile time capsule, the reality is much more complex. Cuban car owners may prize the aesthetic styling of old-school Detroit, but these cars are quite different from the original models under the hood. Bill Visnic, of of the used car website edmunds.com aptly likens the automobile situation in Cuba to the Galapagos Islands where the cars have evolved into something unique.
Sources
"Cuba's Famous Vintage Cars Could Find Their Way Off The Island." http://www.npr.org/2014/12/18/371721089/cubas-famous-vintage-cars-could-find-their-way-off-the-island
Schweid, Richard. Che's Chevrolet, Fidel's Oldsmobile On the Road in Cuba. Chapel Hill: University of North Carolina Press, 2004.
Valdes-Dapena, Peter, "Time travel with Cuba's cars." http://money.cnn.com/2014/12/26/autos/brenda-priddy-cuba-cars/