I'm pretty certain that you can make on one of each type of reservation (dinner, brunch, high tea) on the first day. If you'd like a second dinner later in the week, you can call and see if there are any openings and be placed on the waiting list. You can probably make one for your whole group, and have one of your parents come along with you to make one for them.
As Greg K. said, they do usually start taking reservations earlier than stated in the Navigator, but I've never had to be there a half-hour early. I usually grab some lunch at Topsiders after boarding, and then at 12:55, I run up the stairs to Palo to make reservations, then back down for more shrimp. Although they've usually started by the time I get there, I've never had a problem getting what I want. Note that this is on the Magic, where there are more days and hence more reservation times available. Also, I believe that they take reservations at Wavebands on the Wonder, as opposed to taking them at Palo itself, as they do on the Magic.