no, the alternator of the vehicle got you to the parts store. In a properly working modern vehicle, once the alternator is putting out power, the battery just serves as a buffer, it is not providing power. In gasoline engines, the alternator starts working immediately (some have to reach a certain engine speed); many diesels disable the alternator until they're done using the glow plugs or grid heaters, which are used even after starting until the engine is approaching operating temperature. The OPs car isn't modern, and probably had an undersized generator or alternator even before the additional load of the lights and sirens were added. Also, generator output goes up with engine speed, and low speed parade driving is pretty much the worst case -- all the loads on, engine at or just above idle so generator output is near zero.
@Junkman I'd suggest one of the clore jump-n-carry lead acid packs, probably the 660. You're right that you want a lead acid one, if the goal is run the car on it (and not just start and let the car's charging system handle it, which yours seems not to be able to do) I'd make sure the vehicle battery is fully charged -- using a charger, not the car -- before you go to a parade. AGM batteries have different charging curves and voltages than conventional lead acid batteries, and unless you've adjusted your charging system (which if the car has a mechanical voltage regulator, you could do) it isn't charging it properly. I'd also consider replacing the battery with one with the largest reserve capacity you can get in the right size.
