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.
Place the oil, lemon juice, salt and pepper in a large bowl and whisk to combine. Add the fennel and toss to coat each slice with some of the vinaigrette.
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.