I don't doubt it at all. Take into consideration that he might have served in a local Home Guard unit or something similar for which I do not have records but which you might find in the local court house.
There were two men named A. Paris but I don't think they are your man. One served in the 24th Battalion, Texas State Troops and the other in the 3rd Regiment, European Brigade, Louisiana Militia [Grade Francaise].
There was an Andrew B. Paris who served in the Paris' Comnpany, Virginia Artillery [Staunton Hill Artillery], but again, I don't think that is your man either.
Out of Tennessee there are only the two listed, Albert and Alexander Paris.
JR