Availability of parts is perhaps the largest misconception with regard to the DeLorean automobile. All the mechanical systems on the car (like powerplant and drivetrain) generally use standard parts and can be obtained through typical channels. The DeLorean-specific parts (like body panels and interior components) can still be readily obtained -- the factory stamped out spares for everything on the car before closing down, and you can even buy brand new stainless steel body panels for it still.
There are several shops nationwide that specialize in DeLorean parts and service. The following is a partial list of well-known parts sources.
On occasion you will encounter parts that are becoming hard to find -- usually DeLorean specific trim and components. The front fenders are notoriously hard to find, but the laws of supply and demand have established an equilibrium price for these components as well. Some shops including DeLorean Motor Center engage aftermarket manufacturers for hard to find components and sell reproduction parts that are often better than the originals.