In a mixing bowl combine the beef, rice, 2 eggs, onion, 2 tablespoons of the parsley, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper and mix well. Portion into heaping 1 1/2 to 2 tablespoon portions and, using your hands, shape into balls. Add 6 cups of water to a large saucepan or Dutch oven and add 1 1/2 teaspoons of salt. Bring to a low boil, then add the meatballs, one by one, and simmer until the meat is cooked through, about 10 minutes. Using a slotted spoon, remove the meatballs from the cooking liquid and set aside, covered, to keep warm. Strain the cooking liquid through a fine mesh sieve and return the cooking liquid to the saucepan and keep warm over a low fire.
In a small bowl whisk the egg yolks until smooth. Add the lemon juice and whisk to combine. Slowly add 1 cup of the hot liquid to the egg-lemon mixture, whisking to combine. Slowly whisk this mixture into the remaining hot liquid in the saucepan; do not allow to boil or the sauce will curdle. Cook, stirring constantly, until slightly thickened, 1 to 2 minutes. (The liquid will still be brothy.) Add the remaining 1/4 teaspoon of salt and a pinch of black pepper, return the meatballs to the broth, and serve immediately, in shallow soup bowls, garnished with the remaining parsley.
Note: Meatballs prepared in this manner may be served either as a soup or as a main course. The proportions in this recipe result in a brothy sauce appropriate for soup consistency. However, if a thicker sauce is desired, the sauce may be thickened by addition of 1 or more egg yolks and more lemon juice may be added, to taste, as necessary.