Soak the leeks in warm water for 30 minutes. Drain and cut into 2-inch lengths. Thoroughly rinse to remove any sand. Drain well.
Combine the onion, olive oil, and carrots in a large, heavy casserole and cook over medium heat, stirring, for 2 to 3 minutes. Stir in the leeks and sugar, cover with a sheet of crumpled parchment paper or foil and a tight-fitting lid, reduce the heat to low, and cook for 20 minutes. (Leeks and carrots should cook in their own moisture. If necessary, add 1 tablespoon water.)
Meanwhile, in a small bowl, soak the rice in hot water to cover for 10 minutes; drain.
Add the rice to the leeks. Season with salt and pepper, and cook, tightly covered, over low heat for 20 minutes longer.
Remove from the heat and add the lemon juice. Leave the pan uncovered to cool before transferring the leeks to a serving dish. Spoon the vegetables and rice over the leeks, garnish with the chopped parsley, and serve at room temperature or chilled.