There is also tax added to the price of parking. Have a great stay!
Overnight valet parking is $39 per night. Self-parking overnight is $29 but based on limited availability. There is also a garage behind the hotel, called Mansion House and that is $14 a day without in/out privileges (you will have to pay each time you leave), but if you don't need your car during your stay this is the cheapest. It is not connected to the hotel however.
Get an answer from Hyatt Regency St. Louis at The Arch staff and past guests.