Put the chicken, onion, carrot, parsley, water and salt in a soup pot. Bring slowly to a bare simmer, skim foam from the surface and partially cover.
Simmer gently for 30–40 minutes, until the chicken is tender and reaches at least 165°F / 74°C in its thickest part away from the bone. Lift it onto a plate.
Strain the broth into a clean pot. Measure about 6 cups, topping up with hot water if needed. Add the rice and simmer for 14–18 minutes, until tender.
Pull the chicken from the bones and shred it. Return the meat to the broth and heat through. Taste for salt. Serve with a small sprig of mint if you like; the rice will thicken the soup as it sits.