Heat the olive oil in a large stockpot over medium heat. Add the onion, green bell peppers, celery, and carrots, and cook for 6 to 8 minutes until the onion is soft and translucent, then stir in the garlic and cook 1 minute more.
Nestle the whole chicken into the pot breast side up. Pour in the water and chicken broth, then add the potatoes, tomatoes with their juice, white pepper, black pepper, and salt. Bring to a gentle boil, lower to a simmer, partially cover, and cook for about 75 minutes until the chicken is fully cooked.
Lift the chicken out with tongs and set it in a large bowl to cool for about 15 minutes. Discard the skin and bones, shred the meat into bite-sized pieces, and keep the soup simmering on low.
Lightly mash the softened potatoes and some of the vegetables with a potato masher until the broth thickens and turns slightly cloudy, stopping before it becomes a purée.
Bring a small pot of salted water to a boil, add the ditalini, and cook according to package directions, usually 8 to 9 minutes, until just tender. Drain and set aside.
Return the shredded chicken to the soup, add the parsley, and heat together for 5 minutes. Taste and adjust the salt and white pepper. Serve by spooning ditalini into each bowl, ladling the soup over the top, and finishing with grated Parmesan.