Place chicken into a Dutch oven; pour in broth and bring to a boil over medium-high heat. Reduce the heat, cover partially, and simmer until chicken is no longer pink at the bone and the juices run clear, about 20 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C). Use tongs to transfer to a large bowl. Allow chicken and broth to cool slightly, about 10 minutes.
Remove and discard skin and bones from cooled chicken. Cut meat into bite-sized pieces. Skim fat off the top of the cooled broth and discard.
Return chicken broth to a simmer. Add carrots, celery, and onion; simmer until vegetables soften, about 8 minutes. Stir in chicken, egg noodles, parsley, and lemon juice; simmer until noodles are tender, about 5 minutes. Season with salt and pepper.