Since he was a boy, Senator Ted Cruz has said, all he wanted to do was “fight for liberty” — a yearning that he says was first kindled when he heard his father’s tales of fighting as a rebel leader in Cuba in the 1950s, throwing firebombs, running guns and surviving torture.

Those stories, retold by Mr. Cruz and by his father, Rafael, have hooked Republican audiences and given emotional power to the message that the Texas senator is pushing as a contender for the party’s presidential nomination. In their telling, the father’s experience in Cuba — when the country was swept up by the charismatic young Fidel Castro, only to see him become a repressive Communist dictator — becomes a parable for the son’s nightmarish vision of government overreach under President Obama.

But the family narrative that has provided such inspirational fire to Mr. Cruz’s speeches, debate performances and a recently published memoir is, his father’s Cuban contemporaries say, an embroidered one.