Combine the mustard and vinegar and whisk together. Add the oil in a slow, steady stream, whisking constantly, until combined. Season with salt and pepper to taste. Set aside. Peel and quarter the celery root and julienne on a mandolin or with a sharp knife. Toss with one Tablespoon of the lemon juice in a large serving bowl and set aside.
Core and cut the apple into quarters and julienne. Toss the apple with shaved celery root and the remaining Tablespoon of lemon juice and set aside. Thinly slice the celery stalks on a mandolin and toss with the celery root and apples. Toss with the finely chopped celery leaves, parsley leaves and mustard vinaigrette until well combined.
Top with the toasted walnuts and an additional sprinkle of sea salt if desired before serving.