I don't have a direct answer to your question about where to park for free, but did you consider the distinction between an Annual Pass and a Fun Card? That's kind of the point of an Annual Pass, and it's why Fun Cards are inexpensive. Annual Passes include parking, and they offer other discounts. Unless you are visiting alone, you could buy one Annual Pass for your group and Fun Cards for the others. I know that some tickets can be upgraded, so perhaps you can apply the price you paid toward an Annual Pass. It's worth checking into if you'll be visiting several times this year.
