Bring a medium pot of salted water to a boil and add the beets. Boil until fork tender, 15 to 20 minutes.
Drain and rinse the beets with cold water and, using your fingers, rub the skin off the beets. Chop the beets into quarters and place them in the bowl of a food processor. Add the garlic, pistachios, lemon juice and olive oil and pulse until you have a smooth bright pink spread.
Using a spatula, fold in the Parmesan and season with salt. Set aside.
Bring a large pot of salted water to a boil and cook the spaghetti to al dente, according to package directions.
Rinse the spaghetti with cool water and drain well. Toss the spaghetti with the pesto and serve immediately. Garnish with extra Parmesan and the chopped pistachios.