Girona is a safe city, so you can park outside - just make sure not to leave anything on the sight.

As Barri Vell doesn't allow parking, I usually park around the area north: look for 'Carrer de Palamos, Girona' in google maps, and drive around. Depending on the hour, it may take you a while. Then the apartment is about 10 minutes walk. 

If for any reason you prefer paying for a secure parking, I recomend you Saba Berenguer Carnicer. It is 5 min. walk away, open 24h.