Stir together mayonnaise, lemon juice, and 1/4 teaspoon each of salt and pepper in a large bowl. Cut celery root with slicer into 1/8-inch-thick matchsticks and add to mayonnaise dressing. Cook beans in boiling salted water (1 tablespoon salt for 2 quarts water), uncovered, until just tender, about 4 minutes. Drain, then rinse under cold water to stop cooking. Drain well and pat dry. Toss beans and 1 tablespoon parsley with celery root. Serve sprinkled with remaining tablespoon parsley.