His father had been in the civil war, a captain. The narrator doesn’t specify on which side he fought in the video.
I believe the Hatfields fought for the South, and the McCoys fought for the North.
Cap Hatfield and both sides of the feud were almost entirely Confederate in their allegiance during the Civil War. Afterwards though, the Hatfields were more prosperous and influential based on timbering and were Republicans, while the McCoys were lower middle class Democrats reliant on small scale and subsistence farming and retained Confederate sympathies.