Which political party did the majority of the Mormons belong to in the 1860s?

I don't know of any official statistics, but one can assume that they were Independents. They had been harassed by both Democratic and Republican governments in Washington D.C. and probably wanted nothing to do with either party at that time. Joseph Smith ran for President of the United States under an Independent platform in the 1844 election, and Brigham Young ran for territorial governor of Utah apparently without stating a party and received 100% of the 10,000 votes.

They were Anti-Slavery, Pro-State's Rights, Pro-Women's Sufferage, and deeply rooted in the defence of the Constitution and the Founding Fathers.