Preheat oven to 425 degrees. Toss beet noodles with 1 tablespoon oil; season with salt and pepper. Arrange in an even layer on a rimmed baking sheet and roast until just tender, about 10 minutes.
In a food processor, puree parsley, walnuts, cheese, garlic, remaining 2 tablespoons oil, and 1 to 2 tablespoons water until creamy. Season with salt and pepper.
Transfer noodles to a serving platter. Dollop with parsley pesto and serve, with more grated cheese.