1Place the anchovy paste, shallot, lemon juice, vinegar, and pepper in a small, nonreactive bowl, whisk to combine, and let sit for about 10 minutes. Meanwhile, very thinly slice the celery stalks, celery root, and radishes and place in a large bowl.
2When the anchovy mixture is ready, slowly drizzle in the oil, whisking continuously, until all of the oil is incorporated. Season with salt and additional pepper as needed.
3Pour the dressing over the vegetables and toss to combine. Taste and season with salt and pepper as needed. Transfer the salad to a large serving plate and arrange in an even layer. Sprinkle with the capers, celery leaves (if using), parsley leaves, and Parmesan and serve.