Leftist anti-Christians use the term dominion theology in the same way that, 50 years ago, some right wingers would call anyone to the left of, say, Richard Nixon, a Communist, Communist sympathizer, or Soviet agent. Dominion theology is a subset of conservative Reformed theology that believes that the civil, as well as the moral, law of the Old Testament is binding in the present era, called the church age. The predominant view of conservative Reformed believers is that, while the moral law carries over in the church age, the Old Testament civil law was applicable specifically for national Israel and is not authoritative, but can serve as a guide. Furthermore, the majority view in evangelical Protestantism is dispensationalism, which is prevalent in most conservative Baptist, pentecostal, and independent circles. Dispensationalism differs from Reformed theology in the area of the replacement of national Israel with the church in the present era. Dispensationalists believe that the church is an entity separate from national Israel, with each having different roles in God's plan of salvation, end times events, and the millennial kingdom.
A review of the web site of New Beginnings Church, the Irving, Texas, based church where Ted Cruz's father has preached, indicates the church is not specifically Calvinist. While he may espouse strongly conservative beliefs, there is no indications he is even Reformed, much less a dominion theology advocate. The same liberals who sneer at conservatives calling Obama a closet Muslim or closet Marxist have no qualms about their own name calling, whether racist or dominion theology advocate.
I understand Ted's dad is a dominionist, which I believe is the wrong approach to spreading the gospel. I am also skeptical of Barton. Ted Cruz remains my first choice.