At least 1 hour before you plan to serve the salad, whisk together the olive oil, lemon zest, 1/4 cup of lemon juice, the shallots, celery seed, celery salt, anchovy paste, 2 teaspoons salt, and 1 teaspoon pepper. Place the celery in a mixing bowl and toss it with the remaining 1 tablespoon of lemon juice and 1/2 teaspoon of salt. (Even though these ingredients are in the dressing, believe me?this step makes a difference.) Add enough dressing to moisten well. Cover and refrigerate for at least an hour to allow the celery to crisp and the flavors to develop.
When ready to serve, arrange the celery on a platter, shave the Parmesan onto the celery with a vegetable peeler, then sprinkle with walnuts, parsley leaves, salt, and pepper and serve immediately.