I have done many parades in my A ,all at idling speed, with much use of the horn ----I never charge my battery except by driving
What is your charging rate driving 20-40 mph
How long is the parade, what electric loads do you have during the parade
How old is your battery ---is it fully charged when you start the parade
Overcharging is only a problem after the battery is fully charged, I use quartz halogen bulbs, have my charging rate set for 15 amps ---I use the headlights to prevent overcharging---about 30% of a 100 mile trip I will use the headlights during daylight.
Either you have too much turned on during the parade or you have a weak battery, or it isn't fully charged to begin with ---possibly all 3
Does it have the dual high aux trans ----I would think with the truck axle ratio you would have enough rpm to charge some at a walking speed parade
What are you using for a cutout---perhaps it needs too high of a voltage to "cut in" ---I use a diode cutout, it has given me enough voltage from the generator to hand crank start without a battery.