Rinse the chicken pieces and remove the skin. Place them in a large saucepan. Add the broth and 2 cups of water. Bring to a boil. Reduce heat and simmer gently until the chicken is cooked through, about 35 minutes. Skim any foam that rises to the surface. Transfer the chicken to a plate; let cool. Shred the meat; discard the bones.
Meanwhile, strain the broth and return it to the saucepan. Bring to a boil. Add the carrots, Swiss chard, tortellini, salt, and pepper. Simmer according to the tortellini's package directions. When the tortellini are done, add the chicken and heat for 3 minutes. Ladle the soup into bowls and sprinkle with the Parmesan.