Preheat oven to 350°. Bake walnuts in a single layer in a shallow pan 7 to 9 minutes or until toasted and fragrant, stirring halfway through.
Meanwhile, whisk together olive oil and next 4 ingredients; season with salt and pepper to taste.
Cut beets and carrots into thin slices, using a mandoline or sharp knife. To prevent hands from being stained, hold beets with a paper towel or wear rubber gloves while slicing. Toss arugula with a small amount of vinaigrette. Arrange arugula on a serving platter. Toss beets, carrots, and parsley with desired amount of vinaigrette. Top arugula with beet mixture; sprinkle with walnuts and salt and pepper to taste. Serve immediately with any remaining vinaigrette.