Things are still unsettled since Sandy hit the area, so there may be significant changes from what you can find described on the internet from past years. Belmar and Avon have plenty of municipal parking areas. When I dive the Shark River on a weekday, especially at dusk and at night, I'm usually able to find on the street parking very close to the water. The 'back bay' does have some decent shallow diving, but nothing to compare with the inlet's jetties. The bottom in the back bay is, or was, mostly mussel beds with very little structure.
Shark River is a tidal basin, not a river. There is no river there at all, so there is not as much sediment as might exist at a river site. The best diving is during the period just before and at peak high tide. The amount of marine life can be amazing. In late summer the number of tropical fishes there compares favorably with some similar sites in Florida.
There are no particulary good shore dive sites in Barnegat Bay that I'm aware of, except for the area around the inlet. Because I'm a fish watcher and occasional collector I dive there often, sometimes under the docks used by commercial fishing boats and boat rental operations.
The currents in and around Barnegat inlet can be difficult, and because of the heavy boat traffic there are many restrictions, both legal and practical, on where and when you can dive. This is also true of the Shark River inlet. Weekdays are best. Weekends bring heavy boat traffic and many fishermen. I always carry 3 different line cutters with me because of the monofilament entanglements. The amount of fishing sinkers and fishing lures one can effortlessly collect is amazing.