Yes you can purchase parking tickets separately from show tickets. If you purchase a parking pass in advance we hold your spot all night. You can show up while the headliner is about to finish their encore and your spot will still be waiting for you.
As for filling up quickly, Mongo was right on. Except for the exceptionally bigger shows that cause fans to line up for hours before we open(Timberlake, Manson, Blink) or any show where hundreds of terrified folks from Manassas drive into the ghetto (Train, Dwight Yoakam) you should be safe arriving in the first hour or so after doors open to get a spot. But with the option of buying a parking pass beforehand, why risk missing out?
Otherwise, we dont hear about many cars getting broken into, and as Markie said, if you do park on the street be smart and don't leave valuables in your car, but 7$ aint that much for your peace of mind.