We have sailed out of Southampton on several different cruise lines over the past two years and can recommend CPS, on site at Southampton, for car parking if you are sailing with Princess, P&O or Cunard... these are the cruise lines that CPS deal with. ABP is another on site company that we have used for parking if you are sailing with Celebrity, Fred Olsen, MSC, NCL, Royal Caribbean or Saga. Car parking is bookable online, in advance, with either of these companies.
We always travel down the day before, staying overnight locally. Holiday Inn on Herbert Walker Avenue, Southampton is so convenient and budget friendly but have also splurged on the Southampton Harbour Hotel for a very special occasion!