I do not see anything odd about Confederate veteran marching or riding in parades in the South. It is a shame it did not happen until the 20th Century. I would not expect Confederate veterans would have joined a parade up North. Perhaps in East TN or Kentucky, it might have been appropriate for them to march together.
There is a huge diffence with WWI and WWII . The U.S., all of it, was at war with the Germans and Japs. They should never be invited to a parade in America.