Any car parking bay on city centre streets (not car parks) is free for bikes. Nottingham Council said that they could not police pay and display as someone could remove your ticket from the bike. Therefore, it's all free on the street.
If you want a covered car park, you could try Fletchergate as they have a special bike section. It's not free see the link in Angus's post above.
Up by the castle there is some bike parking with bars to lock your bike to.
There is loads of on street parking around the Lace Market area too.