I know stock tickers were all finally decommissioned in the 1970s in favor of teletype machines (and subsequently, computers)
Who or what had the honor of being the last-ever parade to be festooned with the tape streamers?
The use of actual ticker tape has, as you suggested, declined since the computerization of Wall Street. However, at least as late as 1991, some ticker tape was being used in a parade. Trans-Lux, then one of the last manufacturers of stock tickers and the associated printing material, provided about 500 pounds of ticker tape out of some 12,000 pounds of overall material used when troops came home from Desert Storm. I've found some indications that it may have been used since then, but nothing for which I can find good resources.
Source: "New York Salutes; A Lot of Stuff, A Little of It Ticker Tape", New York Times, 11 Jun 1991
How did ticker tape begin as a thing for celebration?