The KKK terrorized and murdered Republicans.
At the time, for obvious reasons, all Negros were presumed to be Republicans and that is why the Democrats murdered blacks more often than white Republicans. The dark color of their skin was proof enough to identify Negros as Republicans. More proof was needed to prove that a white man was a Republican, therefore the Democrats murdered blacks at a greater rate than they murdered whites.
Those elements - inside and outside the Klan, before and during its existence -- were already murdering and coercing blacks. The Klan once it ran amok certainly persecuted people who were Republicans; they did the same on people who happened to be federal personnel and who happened to be "carpetbaggers" as well as the occasional philandering husband or debt deadbeat. That doesn't make them bankers, priests or communists by some fallacy of exception.
What's missing here is this cockamamie idea that all elements of everything that happens in human history can be reduced to one of two elements, either "Democrat" or "Republican". That's nowhere near the world of reality, which is far more complex than that. Those targeted black ex-slaves for instance, were targeted for trying to exercise the right to vote --- but you left out that they were also targeted for having the temerity (<sarc) to walk into town in public, or attempt to conduct business with whites, or apply for jobs or expect to be paid for them.
Again, the nefarious activities practiced by the Klan were going on long before they existed, and continued long after they were extinguished. And then they bubbled up in the second one. The Klan was one of the symptoms of it. But to try to reduce it to some binary formula of 'one of two political parties' is cherrypicking nonsense.
Another factor here is that its original incarnation existed only in the South --- yet another demonstration that it was a cultural expression. Had it been part of a political party it would have existed wherever that political party did, which was all over the nation. But it didn't -- because it was derived from a Southern cultural background. "Slave patrols" had been around since 1704. All that was new in that vein was the name and the attire.
And again ---these same elements that did similar Klanlike things, existed in dozens of different groups in various regions all (again) in the South. I've listed some of them earlier. We single out the Klan in our time because it was made into a movie and then it was restarted by an opportunist taking advantage of the bigotry of his time, but in its own era the original Klan was one of literally dozens of similar groups popping up all over the vanquished Confederacy -- as well as ad hoc vigilante posses who never took the time to organize into a formal name. What about these groups? Were they all founded by a political party too? Doesn't seem real efficient.
This is a little like singling out a Hyundai in a full parking lot and going "see? Korea invented cars".
Keep on lying to yourself.