Why don't you just get in contact with Royal Caribbean and you will be able to park your car in exactly the same way as those with "free" parking, except that you will have to pay? A specialist parking agency deal with all the cruise parking in the port. Happy cruising.
You can book direct with at www.abports.co.uk (the car park is next to the ship and you only have to walk approx 100 yards).
You might also like to consider comparing the cost of fuel and parking with the cost of getting a taxi down and back. I would speculate that a taxi will cost about £100 return and takes all the strain away and lets you start your holiday earlier. As an afterthought recently the M3 has been slow on the approach to the M27 because of major road works on the M27 towards the signposted docks route. Turning towards Portsmouth and taking the next exit and then going into Southampton will alleviate the situation somewhat but queues on Saturdays have reached as far back as Winchester ... Good Luck ... Neil.
If you have a look at page 120 of RCI 2009/2110 brochure, it states that free parking is available at So'ton to all passengers of Independence of the Seas, except Trans Atlantic crossings, (1 space per booking), as long as bookings are confirmed by 31/12/2008. A lot of TA's I have spoken to are trying to make it look as though they are giving you something for nothing. Alas they are not, RCI are , but probably hiding the parking charge somewhere in their price.
As you did not say when you are sailing this may not apply to you ,but check it out anyway. Bon voyage!!