Welcome to planDisney, Amy! I'm so glad to be able to answer your question today.
I have some good news for you: since you purchased date-based tickets, you do not need to make Theme Park reservations. On each day of your trip, you'll just head to the park you want to visit and scan in, no advance assignment to a specific park required. In
My Disney Experience, you only assign tickets to people (make sure each ticket shows the correct guest in Tickets & Passes); you don't need to assign them to parks.
About that "Magic Kingdom not available" message you're seeing for December 31: that notice applies to admission types that still require reservations, like Annual Passes, not to date-based tickets like yours. It doesn't mean Magic Kingdom is already full for the day or that you can't enter with your tickets!
That said, a friendly heads up: New Year's Eve at
Magic Kingdom is extremely popular. That park often reaches capacity at some point on December 31, and may temporarily restrict entry. The best plan if Magic Kingdom is your must-do is to arrive early and be prepared to stay for the day. You can leave and come back if the park hasn't hit a closure phase, but once capacity restrictions kick in, re-entry isn't guaranteed, so it's smartest to plan meals, breaks, and fireworks viewing all inside the park.
I hope this answers your question, Amy, and please let us know if there's anything else we can do to help you plan your magical vacation!
Jessica