Shred the potatoes, carrots, and parsnips by hand or in a food processor. Spread the vegetables on a clean dish towel, wring out any excess moisture until the vegetables are thoroughly dry. Place in a large bowl.
Blend flour, dill, green onions, salt, and pepper together in a small bowl. Add this mixture to the shredded vegetables mixing well. Mix in beaten eggs.
Heat ¼ inch of oil in a skillet over medium heat. Working in batches, drop 2 heaping tablespoons batter per latke into the hot oil. Flatten with a spatula until the latke is between 3-4 inches. Fry until brown on each side, about 3 minutes per side.
Remove from skillet and place on paper towels to absorb any excess oil.