What do you mean by the Vatican? Most people want to see St Peter's Basilica, the Vatican museums and the Sistine Chapel. The queue at the Basilica is mainly for the security checkpoint, but the queues for the museums can be pretty dreadful, an hour is conservative at this time of year. The Sistine Chapel is at the far end of the museums and there really isn't way of short circuiting this. You can cut down on the queuing by buying tickets in advance, online, or joining one of the guides that you'll find outside the museums. They get the tickets without queuing, and it might be worth it just for this.You don't actually have to take the tour if you don't want to. One way or another, you're still talking about most of the day to do it all justice. Remember that the Vatican and the Basilica have dress rules too, for men and women, and they do enforce them.
Can't help you with parking. I've never really fancied driving in Rome..!