We went to WDW for the biggie New Years (2000). My DD16 and her friend got up early and went to Magic Kingdom and we had intended to sleep in (flew in late the night before) and then join them (we, being my husband, myself and my DD7). Well, by 11am the gates to MK were closed. We kept in touch with her by cell phone and she said they were only letting in resort guests, which we were not.
We decided we would just hang out at the hotel and enjoy the day at the pool. Later on we found out through some guests at our hotel that Animal Kingdom wasn't at capacity. We drove over there about 6pm and had no problem getting in. We had an absolute great time! They couldn't do the fireworks in the park because of the animals so they did them in the parking lot. It was quite the show and we could see the fireworks in the other parks from a distance. We thoroughly enjoyed this.
DD16 and friend ended up staying at Magic Kingdom that day from 10am to 2am! We stayed in touch all day and they had a great time also. Something she will always remember, although we were all a little sad that we weren't together.
I can't say for sure, but based on our experience, you should be able to get into one of the parks later in the day. Typically, MK fills first, then MGM, then Epcot (this is a huge park that doesn't fill up early in the day) and lastly Animal Kingdom.
HTH
Jenn