That's false. Rubio's parents came to the US before then, in 1956. They remained in the US after Castro took over in 1959. They returned to Cuba for brief stints early on, before the country devolved into Soviet-style totalitarianism.
But the top of the story suggests Rubio himself has given this "dramatic account:" that "he was the son of exiles, he told audiences, Cuban Americans forced off their beloved island after 'a thug,' Fidel Castro, took power." (Update note: The story struck the word "dramatic").
However, the story doesn't cite one speech where Rubio actually said that.
To back up the lead, the Washington Post excerpts from a 2006 address in the Florida House where Rubio said “in January of 1959 a thug named Fidel Castro took power in Cuba and countless Cubans were forced to flee... Today your children and grandchildren are the secretary of commerce of the United States and multiple members of Congress...and soon, even speaker of the Florida House.”
The catch: If you listen to the speech, Rubio isn't just talking about those who specifically fled Cuba after Castro took power. He doesn't say that his parents fled Cuba. Instead, he was talking about "a community of exiles." That is: He was talking about all the Cubans who live in Miami.
Regardless of when his parents left Cuba, they were exiles because they stayed in the US, specifically Miami, in a community where they soon felt they couldn't go back to their homeland. Though the story said his parents left for economic reasons, it's silent about the fact that the dictator before Castro, Batista, was so brutal that it made Castro look like a good alternative at first. (Insert debate over the fairness of the post-Castro Cuban Adjustment Act here).