Cut off the leafy tops of the beets, leaving about 1/2-inch of the stem, and reserve them for another use. Wash the beets well to remove all of the sand. Peel with a vegetable peeler while wearing gloves.
Cover the peeled beets with cold water in a medium saucepan and salt generously. Bring to a boil over high heat, then lower the heat to medium and cook until fork-tender, 20 to 30 minutes. Drain the beets into a colander and cut them into quarters.
Meanwhile, combine the parsley, walnuts and garlic in a food processor and pulse until coarsely chopped. Add the olive oil and process until smooth. Stir in the Pecorino and vinegar and season with salt.
Toss the beets in a bowl with some of the pesto. Transfer the remaining pesto to a small ramekin and serve the beets with more pesto on top.
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