In a bowl, whisk olive oil, mustard, vinegar, and 1/4 teaspoon salt to blend.
In a 4- to 6-quart pan, bring 3 to 4 quarts water to a boil. Meanwhile, peel celery root and cut into 2-inch-long matchstick-size strips. Add celery root to water and cook until tender-crisp, 2 to 3 minutes. Drain and immediately add to dressing, along with parsley. Mix to coat. Add salt and pepper to taste. Serve warm or at room temperature.