They do a christmas parade and it is taped not live....usually the first weekend of Dec. They do perform this parade everyday at MK in some variation....
Disney does not do a special Thanksgiving Parade, but they do have a Christmas Parade. The version that airs on Christmas is actually taped much earlier in the month, with some segments shot live on Christmas Day.
Just wanted to add that that is Everyday AFTER December 17. The Christmas Parade is exclusive to the Very Merry Christmas Party until then. After the last party (Dec 17) it will become the Day parade during regular park hours (and usually starts at 2pm vs the typical 3pm day parade start time as it is a fairly long parade).