Hey, I was stuck as one of those people who reserved one day and was given the "no parks available" message. The park I reserved was for the first day. On, a hunch I cancelled that first day park reservation and...it let me make the reservations for the rest of my stay. The key (for me at least) was to cancel the first day reservation , make the other days starting from last day, and make the reservation for my FIRST day LAST. I was able reserve all the days of my trip this way. I tested this a few times and it worked each time.
So those desperate with "one park reservation and the rest of their trip has no parks unavailable", this might be worth a shot?