Sen. Ted Cruz's father immigrated to the United States from Cuba. (Justin Sullivan/Getty Images) Sen. Ted Cruz (Justin Sullivan /Getty Images)

The father of Sen. Ted Cruz (R-Tex.) said in an interview airing Thursday that he had to grease some palms -- in Cuba, at least -- to get to the United States.

"A friend of the family -- a lawyer friend of my father -- basically bribed a Batista official to stamp my passport with an exit permit," 74-year-old Rafael Bienvenido Cruz told NPR.

The elder Cruz said he fought alongside Fidel Castro during the Cuban Revolution against dictator Fulgencio Batista and that he was captured by Batista's forces.

He emphasized that he went through all legal channels while applying for a student visa from the U.S. government to attend the University of Texas.

"I came to this country legally," he said. "I came here with a legal visa, and ... every step of the way, I have been here legally."

But he said the exit permit was the one thing standing in his way. "Then the only other thing that I needed was an exit permit from the Batista government," Cruz said.

The senator has said that his father's immigration experience has shaped his views on the issue. Cruz has emerged as one of the chief critics of the bipartisan Senate immigration reform bill and has introduced amendments to the bill, including one that would strip the legislation's path to citizenship for undocumented immigrants.