I don't think so, as the hotel is very central and opposite the tram. But better to ask management. I would not take a car into Zurich downtown as public transport is excellent and frequent. I saw some underground parking near the opera house, about 20min walking from the hotel. I am sure there may be others in downtown.
No specified parking that I was aware of. The person at the desk directed a man in front of us to a paid parking lot/garage a few blocks away. Sorry I can't be more help.
No parking, according to what I saw. I believe near the railway station there might be one.
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