What is the deal with Ted Cruz' father? I've read several accounts

[Bucky]

2. Cruz was born in Canada.

From what I understand, two parents were involved. The Wiki you don't trust (could you try Conservapedia instead?) says his father left Cuba in 1957. That's not someone who "escaped from Cuba". He might've left for the violence, but if he left that early for political reasons, he was an anti-Batista partisan.

Batista was a socialist. There was a non-communist guerrilla campaign as well as Castro's vaguely leftist moement. In 1957 the pro-democratic rebels got clobbered by the Cuban army, so if Cruz's dad left in that year, he might have a strong claim to anticommunism. Castro's forces didn't drive out Batista until a year and a half after that--and it was only after taking over the country that Castro announced he was redder than a baboon's ass.

It's plausible that Cruz senior had to run away from Cuba, but the timeline shows it wasn't communism he ran from.