Parking at any downtown hotel is expensive which is why I usually recommend that visitors not bring a car if possible. Keep in mind that the the hotel will usually offer in out privileges so depending on how long you park at any destination you may end up paying at least as much as you would pay at your hotel. There are a couple of public parking garages in the south of market area that will run you between $25-35 for 24 hours The two that are closest to the Palace hotel are on third between Howard and Folsom across from Moscone Center or at 5th and Mission. You may also be able to find over night parking for $15 at some of the lots south of market.
It is a few blocks to walk but parking at the 5th MIssion garage is about $15 a day.