I'm not quite I understand what conservatives are doing when they bring this kind of thing up:
The original targets of the Ku Klux Klan were Republicans, both black and white, according to a new television program and book, which describe how the Democrats started the KKK and for decades harassed the GOP with lynchings and threats.
Which is precisely the point. Decades ago, the Democratic Party was, among other things, the political home of white supremacy in the United States. In the 1960s, the party's leadership decisively broke with that record. At around the same time, part of the rise of the conservative movement inside the Republican Party was the growing prominence of folks like Barry Goldwater who opposed the Civil Rights Act and who found in his 1964 campaign that the main electoral constituency for his brand of conservatism was . . . white supremacists. Other white supremacist politicians (some of whom, unlike Goldwater, would forever remain unrepentant) like Strom Thurmond and Jesse Helms moved into the GOP column. And of course while explicit advocacy of segregation has long since vanished from the top ranks of the Republican Party, major conservative leaders have been heard in recent years issuing paens to the work of Thurmond and Helms, with key legislative leaders specifically regretting that Thurmond's 1948 white supremacist presidential campaign failed, and pointing to Helms as exemplifying what conservatism is all about.
But, yes, decades ago things were different.