I think I read somewhere that in countries with lower levels of gender inequality (like ours), men are more authoritarian than women because they feel more threatened by the increased competition. And with our tortured history of race relations, that is especially prevalent among white men.
It is the other way around in countries and societies that are still blatantly patriarchal, like Saudi Arabia.
Saudi Arabia is an absolute monarchy. That's as authoritarian as you can get.
http://www.dnaindia.com/lifestyle/report-women-more-authoritarian-than-men-study-1710117
I had read somewhere that in countries with higher gender equality, men held less authoritarian views due to the social stigma attached to it. And in places with lower gender equality, women were more likely to support authoritarian views and values such as obedience as it would allow them to survive in that culture.
To the OP, the GOP clearly holds more authoritarian views than the Democratic Party. Social conservatism and authoritarianism often go hand in hand.