Even if the specific hotel doesn't have free parking, you should look into whether there's safe street parking somewhere nearby. I've stayed at lots of hotels near Disneyland (not going there, just because they're cheaper in that area), which charge $15 or more for parking, but I just park in a residential area two blocks away for free (just watching the "no parking Someday at these hours due street cleaning" signs). So just because a hotel charges for parking doesn't mean there can't be free parking very close by.

But since you don't want to name the hotel, we can't offer nearby parking suggestions for you. So perhaps do what I do (when I don't have experience in the area already or crowd-sourced suggestions), use Google Maps to examine the nearby area. If you can get street view, you may be able to read the parking signs! Or look to see if you see smaller residential street nearby.

Gosh, if you can find free parking a mile or two away even, as long as its within the range of the hotel shuttle (if the hotel shuttle isn't limited to the airport), you could just call from wherever you park and ask for the shuttle to pick you up there. (You'd just need to find the closest intersection to get the street names, either visually or an a map on your phone.)