In NY - and I'm sure other states due to the federal nature of CDL, FMCSA, DoT laws etc., rotating beacons/flashing lights are to be used in recovery operations by stationary wreckers - at scene of accident etc., but must be turned off when joining the traffic again - with or without vehicle in tow... UNLESS said vehicle in tow is over width or slow moving (as well as having remote trailer lights). Equally, oversize loads up to a certain weight need solid amber rear lights or flashing if over that limit.
However - as far as 'tow trucks' or pickups with tow dollies, most people seem to get away without trailer lights if useing the beacons.
They are really to advise other road users that you are a hazard, above and beyond your usual lights advising of your presence or intentions...
The rest of NATO calls convoy lights, the 'blackout' lights that only illuminate a white patch of paint, under the rear of the vehicle or trailer, painted diff etc., so only visible by a close follower and the feable little lamps up front.
I think as far as parades go, "anything goes", even Stars n bars tied to a horses butt, (that's another story). It would take a really miffed trooper who was DoT certified to have a prob with amber beacons in regular use.....