You can also look at Park Whiz for locations near the Palmer. Cheaper and you know you have a space.
I work downtown and take the train but I do know that a variety of parking options are available for your convenience such as overnight self parking, VIP Valet Parking and Assisted Valet Parking. You can check their website for pricing.
$45 if you go through the Palmer House
$48 at hotel per day sounds right there's plenty of cheaper self parking lots nearby with rates as low as $12 a day
$48 Per Day
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