Preheat oven to 350 degrees. Place pecans on a baking sheet, and toast until fragrant, about 10 minutes. Transfer to a bowl to cool. Coarsely chop, and set aside. Raise oven temperature to 450 degrees.
Trim greens and long roots from beets. Wrap in 2 to 3 aluminum-foil packets, dividing beets according to size. Place in oven, and roast until tender, 60 minutes or longer, depending on size of beets. Using paper towels to protect your hands, wipe the skins from the beets. Cut into ½-inch pieces, and transfer to a serving bowl.
Drizzle vinegar and olive oil over beets; toss to coat (beets can be made ahead). When ready to serve, add feta, parsley, pecans, and salt and pepper to taste; toss to combine. Garnish with parsley sprigs.
