I notice the hotel advises patrons to use the 'manned' multi-storey car park at Chorlton Street , probably your best bet .... at least your car should be there in the morning !
There is another multi-story a few hundred yards away too .... on Whitworth Street.
There are a few side streets opposite the hotel where you could probably take a chance and park , but it's completely at ones own risk ....... as are most of the car parks in the immediate vicinity of the hotel , some are no more than wasteland ...... the area around the Princess is busy and very tight.
There's a few of these 'car parks' around the gay village area (two minutes walk away) if you want to take a chance !