In a large salad bowl, combine the olive oil and vinegar, and mix well with a fork. Season with salt and pepper; set aside.
Add the celery to the bowl with the dressing and toss to combine. Remove celery with a slotted spoon; set aside.
To the bowl with the dressing, add the lettuce and parsley. Toss to coat with dressing. Arrange the greens on a platter, and top with celery. Using a vegetable peeler, or a paring knife, scrape the cheese to make thin sheets and layer them over the celery. Serve immediately.