Combine the chicken broth, celery and all but 1/2 cup carrots in a large pot. Cover and bring to a simmer over medium-high heat. Reduce the heat to medium low and add the chicken, 3/4 teaspoon salt, and pepper to taste. Cover and gently simmer, flipping the chicken once, until just cooked through, about 18 minutes. Remove the chicken to a cutting board; let cool slightly. Reserve the broth.
Meanwhile, combine 4 cups water, the rice, tomato paste, 1 chopped tomato and the remaining 1/2 cup chopped carrots in a saucepan. Cover and bring to a boil, then reduce the heat to low; simmer, covered, until the rice and carrots are tender, 15 minutes.
Puree the rice mixture in a blender until smooth; whisk into the pot with the broth mixture. Add the peas, heavy cream and the remaining 2 chopped tomatoes. Bring to a simmer.
Shred the chicken and return it to the soup. Stir in the parsley and lemon juice; thin with water if needed. Season with salt and pepper.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.