Fill a medium bowl two-thirds full with cold water; add 1/4 cup lemon juice. Trim top and bottom of celery roots, then peel with a vegetable peeler or paring knife. Transfer to lemon water.
Toast walnuts in 1 tablespoon oil in a small skillet over medium heat until fragrant, 2 to 3 minutes. Add shallot; cook, stirring, until soft and translucent, about 2 minutes. Stir in lemon zest. Set aside.
Whisk together remaining 1/2 cup lemon juice, vinegar, and remaining 3 tablespoons oil in a small bowl. Cut celery root into matchsticks (or julienne using a mandoline). Toss celery root with walnut mixture in a serving bowl; add parsley. Pour dressing over salad. Season with salt and pepper; toss well. Let stand 15 minutes before serving.