Preheat the oven to its lowest setting. Using either the largest holes of a hand-grater or the shredding blade of a food processor, grate the potatoes, onion, and carrot. Place the mixture in a sieve and press to remove as much liquid as possible.
Combine the potato mixture in a large bowl with the eggs, parsley, salt, and pepper. Using your hands or a wooden spoon mix well. Stir in 1/2 of the bread crumbs, then add the remainder slowly, stopping when the mixture is just thick enough to hold its shape, but not so loose that it dribbles off a spoon.
Pour oil to a depth of 1/4-inch into a large, heavy skillet and heat over medium heat until a tiny drop of the batter sizzles upon contact. Using a 1/3-cup metal measure as a scoop, ladle the batter into the hot oil in the skillet, pressing each mound of batter into a small flat pancake: do not crowd the skillet. Fry the pancakes until nicely browned on both sides, 4 to 5 minutes per side.
Transfer the pancakes to a platter and keep warm in the oven while you fry the remaining pancakes. Serve immediately.