Place the chicken, carrots, celery and onion in a large pot. Add the broth, cover and bring to a boil. Reduce heat, add ¾ tsp salt and ½ tsp pepper and simmer, covered, until the chicken is cooked through and the vegetables are tender, 15 to 20 minutes. (Skim and discard any foam that rises to the top.) Meanwhile, cook the rice according to package directions. Transfer the chicken to a bowl. When cool enough to handle, shred the meat, discarding the skin and bones. Stir the chicken back into the soup and bring to a gentle simmer. In a medium bowl, whisk together the eggs and ¼ tsp salt until frothy. Whisk in the lemon juice. Whisking constantly, gradually add 1 cup of the hot chicken soup (liquid only) to the egg mixture. Gradually whisk in an additional cup of soup. Stir in the parsley. Stir the egg-soup mixture and rice into the soup in the pot; warm for 2 minutes (do not let it simmer or boil).