If the fennel still has its stalks, trim them away. Cut the bulbs into quarters, top to bottom, and remove the core and any unattractive outer leaves. Cut the fennel quarters against the grain into paper-thin slices.
Mound the salad attractively on a platter or divide among 4 plates. Shave the Parmesan over the salad using a mandoline or truffle shaver. Sprinkle with parsley and serve.