"Wow, you must really hate those Democrats, right?"

They have found a smug lie that they can use as an excuse to vote for the bigotry that exists on their side of whatever wedge issue drives their entire identity. Or it is latent racism they don't want to consciously admit to. Probably both.

Nothing you say or do will get them to admit Repubs are campaigning on a platform of bigotry. "Southern Strategy" "Nuh-uh." "Goldwater literally campaigned on voting against civil rights." "Nuh-uh." "Why are all the Confederate and Nazi flags welcome at Repub events?" "Nuh-uh." "Donald called all Mexicans rapists and thieves." "That's about the economy."

Source: was slamming my head against this very brick wall last weekend.

I think the easiest solution, if the person you're talking to is not a conservative and is just the embodiment of the adage "A little knowledge is a dangerous thing", is to point out that before, during, and immediately after the Civil War, the Democrats were the conservative party, and the Republicans were liberal - to the point of radicalism, even. The Republican Party of the 1850s and 60s was a VERY big tent consisting of just about everyone who opposed slavery - from ultra-wealthy industrialists to LITERAL Marxists. But nowadays, the Republican party quite openly advertises itself as conservative.

The people have the same agenda - or at least similar ones. Only party preference has changed.

Otherwise you could point to the Southern Strategy leading the GOP to court racist conservatives, the same demographic who caused the Civil War; or that almost all major civil rights legislation post-Grant have been under Democrat administrations. Brown vs. Board of Ed was during Eisenhower's term, but Ike:

A. Was asked to run by both parties

B. Disdained the conservative wing of the GOP

and

C. Was done by the Supreme Court, which Eisenhower had no (immediate or direct) say in.

Yes, and the South all voted Democrat. Things change.