I've never known them to check guests credentials once you are on resort property, like for instance you are going towards the beach area to sit and watch the fireworks. You already have a legitimate 'Ohana reservation for dinner so you should be able to park your car. If you are parking, they usually say a 3 hour limit for dining, but also, I personally have not seen them enforce this. If you are worried about it, you can valet park your car for the day for $25 and you can spend your evening at the Polynesian resort without worrying about your car.
If you are taking a bus from somewhere else on Disney property to get to your dinner reservation, then you don't have to worry about your car.
If you are worried about watching from the beach, you can also view fireworks from the Poly boat dock/pier.
