On the 7-nights, there are four "themed" nights, where everyone has the same dress code and eats the same menu regardless of which restaurant they're in, and three non-themed nights where you eat your restaurant's menu. So the answer to your question is that on formal or semi-formal nights, the dress code is the same in all restaurants, and yes, you could be in Parrot Cay for one (which is really no big deal). The other nights are a Tropical/Mexican night and a Small World night (which I think has been re-named). On these nights, the menus are the same in all three restaurants, too, but dress is casual.
Dress for Palo is the same every night. It's not really formal, but jackets are recommended, and it's a bit more dressy than the regular dining rooms, but not as dressy as the formal nights.
Hope this made sense.....