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Speaking by telephone with "Fox and Friends" on Tuesday, the Republican presidential candidate took issue with Cruz's father recently suggesting to an audience of Christian voters in Indiana that Trump's hypothetical presidency "could be the destruction of America."
"I think it's a disgrace that he's allowed to do it, I think that it's a disgrace that he's allowed to say it," Trump said of Cruz.
He continued: "It's disgraceful that his father can go out there and do that, and so many people are angry."
Trump also pushed back against the elder Cruz's pitch to evangelical voters, pointing out high-profile endorsements from figures like Liberty University President Jerry Falwell Jr.
During Tuesday's interview, Trump also sought to undermine Rafael Cruz's legitimacy by parroting a dubious National Enquirer story. The story claimed Rafael Cruz was photographed in the early 1960s handing out pro-Fidel Castro leaflets with President John F. Kennedy's assassin, Lee Harvey Oswald.
Various news outlets have cast doubt on the legitimacy of the Enquirer, which has endorsed Trump and has published a series of suspect stories critical of the Texas senator.
Trump's comments came the day of the Indiana primary, in which the Republican frontrunner is looking to score a knockout blow against Cruz. Though Cruz has campaigned hard in the Hoosier State, most recent polls show Trump with a significant lead.