The city council have worked at removing free car parking to create parking for residents instead, so the city centre mainly has huge car parks with fees.
The shopping mall Mobilia to the south of the centre has free parking for their guests, but only for three hours or so (although I've never seen any parking guards) and there you can catch buses to the centre. The Caroli City shopping centre not that far from the central station, has free parking Saturdays and Sundays I think, but I'm not sure if they are open that long and they are currently rebuilding parts of the centre.
The street called Regementsgatan, close to the main library, has some spaces I think, as does the streets along the Pildammsparken park but you would have to check which ones since I don't drive so I just see cars parked there as I cycle along to take my daughter to football.
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