Question

Why did Cuba partner with the Soviet Union during the Cuban missile crisis

Answer 1

Cuba as well as the Soviet Union had a common interest. Which was to promote and expand communism. Cuba tried to promote communism to the Americas and the Soviets had Cuba as one of their most closest allied to the United States. 

Answer 2

*NOTE* Before you read below, Notice that I AM NOT taking credit for this answer, I just found it and thought it could help:
The Cuban Missile Crisis. ... The Soviet Union had secretly stationed nuclear weapons on the island of Cuba, and when the government of the United States discovered them, and demanded their withdrawal, the most dangerous confrontation of the Cold War followed.