The parks might give you a "Happy Birthday" pin. I was there for my birthday with my parents and my friend. We didn't go to the park(s) on my actual birthday, but we did go to eat at Bubba Gump's. They do announce that it's your birthday to the restaurant, and you get a sundae.
If you stay on-site, let the front desk know it's your birthday. They'll send up a gift basket and a personalized card to your room!