If Captain Parmenter was transferred to Fort Courage immediately after he "led" the cavalry charge that led to the Confederate surrender at Appomatox Court House, then Trooper Hanibal Dobbs should not have been there. Immediately after the Civil War, Dobbs, being a Southerner, would have been ineligible for service in the United States Army. Lee's surrender at Appomatox did not, in fact, end the Civil War. The war would not end for another ten weeks. So, it is possible that when newly-minted Captain Parmenter transferred to Fort Courage, the Civil War was, in fact, still going on.
The troopers carry Springfield carbines and Colt Single Action Army revolvers. Both firearms were adopted by the Army in 1873, several years after the show's time frame. Also, the skinny barrels of the Colts would suggest a .22, not the .45 actually used.
When Capt. Parmenter asks why Reveille is three hours later than it was when he was back east, Sgt. O'Rourke responds by saying it's the difference in the time zones. There were no time zones in the Civil War era as there was no need for them, since communities kept their own local time. It wasn't until the advent of transcontinental railroads, and the need to maintain a standard regulated time across the U.S., that time zones were instituted in the late 1880s.
Wilton Parmenter is given a Medal of Honor then a Purple Heart for being wounded with the pin. The Purple Heart of the modern design shown being pinned on Parmenter was not created until 1932. Its predecessor, the Badge of Military Merit, had a different design and was not awarded again after the Revolutionary War, although not officially abolished.