Bring chicken broth to boil in large pot over medium-high heat. Mix in carrots, celery, onion, then rice; return to boil. Reduce heat to medium-low, cover, and simmer until rice and vegetables are tender, about 20 minutes. Remove from heat. Ladle 2 cups hot broth from soup into measuring cup. Using electric mixer, beat whites in large bowl until stiff but not dry. Beat yolks into whites. At low speed, beat lemon juice, then 2 cups hot broth into egg mixture; return mixture to soup. Rewarm soup (do not boil). Season with salt and pepper. Ladle into bowls. Top with lemon slices and parsley, if desired.