The closest large car park to the Palace Theatre is the Palace NCP car park on Whitworth Street (use the postcode M1 6NE). It costs £11.50 to park here for 3 hours, and it's open 24 hours a day. It takes less than 5 minutes to walk to the theatre from here at buggy pace.
There are cheaper on-street parking options within a short walk of the theatre, but please note that spaces are very limited. You can find all other parking options near the venue on the Parkopedia website.