We've been to WDW twice on NYE, this past year and 1998. We stayed on site both times and had an information sheet put under our door both times to "warn" us that we would be able to get into a park on NYE, but it might not be the park of our choice, especially if we went after noon or if we park hopped.
From what I've figured out/been told by CMs and other guests, the hierarcy of what happens when it's full goes like this:
-when a parking lots get full, it closes and that park is closed to guests coming by car
-other parking lots that have room remain open and people coming to the closed lots are re-routed to another park
-the parks have more capacity than the parking lots do, so even though parking is closed, the park might be still open if you are using WDW transportation or walking
-when a park becomes full, WDW transportation stops taking people to that park and takes tehm somewhere else instead (we talked to some people who got on a bus going to MK and during their ride, the destination was changed to Epcot)
-when a park is really full, access to it may be denied even to people who are walking there
-as people come and go during the say, the parking lot or park might re-open
-there was also something about park hoppers in the mix, but I'm not sure how to explain it. If they are limiting park access, if you have already been in a park that day, you would get in before someone who has not yet entered a park (I guess, the idea is that someone who has already used a day of their pass gets in before someone who has not used their pass yet).
We were told that Epcot has the most capacity and was least likey to close.