Contact www.abparking.co.uk - they have the carpark right next to the doc.
We used www.cruiseparking.co.uk when sailing from the QE2 terminal, also next door.
We used CPS, bookable through Cunard. We drove into the terminal, the porters unloaded our luggage and then they took our car to a secure car park. Unpon our return, we gave them our slip and they brought our car to the terminal building. Very easy.
Try PPS, Southampton they are good. Have used them several times.
Excellent service used on several occasions would like to pay less but the good chappies have not given me areduction yet.