Bring chicken, water, and chicken stock to a simmer in a large saucepan over medium-high heat. Reduce the heat to medium-low and continue to simmer until chicken is no longer pink in the center, about 20 minutes. Remove chicken and set aside to cool. Reserve the cooking liquid.

While the chicken is cooling, heat olive oil in a large pot over medium heat. Add onion and garlic; cook and stir until onion has softened and turned translucent, about 5 minutes. Add potatoes, then strain the reserved cooking liquid into the pot. Increase the heat to high and bring to a boil, then reduce the heat to medium-low and simmer until potatoes are tender, about 25 minutes.

Cut cooked chicken into cubes and add to simmering potatoes; cook for 5 minutes. Stir in garbanzo beans, spinach, and roasted red peppers; simmer for 10 more minutes. Season with salt and pepper. Sprinkle with Parmesan before serving.