Slavery did not last until the Cuban Revolution. It was legally abolished in the 1880s and had long since been ended in all forms by the 1950s.

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Rural laborers were still exploited in various ways, of course, such as the colono system where big sugar cane mills would charge low prices for sugar from smallholders and sharecroppers, but this was different from slavery.

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I am guessing that the misconception here comes from the meme online that 'lol you are just mad that Fidel expropriated your family's slave plantation'. The meme started as an overstatement and I think that while the creators knew it was half jokingly calling large landholders slave owners, I don't think a lot of the people who come across those memes in the wild always get that it's meant to be partially joking. At most I've seen people defend it as 'wage slavery', but that is different from just calling it 'slavery' since normally that implies *chattel slavery*.

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It is true that the first big wave of post-1959 Cuban immigration was made up largely of middle class and wealthy Cubans, including many big landowners, but they were not slave owners.

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For a summary history of Cuban immigration to the US, you can read Felix Masud-Piloto's *From Welcomed Exiles to Illegal Immigrants: Cuban Migration to the U.S., 1959-1995*. For a broader history of the Cuban Revolution that covers this kind of thing, check out Aviva Chomsky's *A History of the Cuban Revolution*. I'm also more than happy to make even more specialized bibliography recommendations but for the purposes of this specific question I don't think it's necessary because the idea that slavery survived into the 1950s is patently false.