Heat olive oil in a large stock pot over medium heat; saute onion, carrot, celery, and garlic until tender and slightly browned, about 5 minutes. Add water, ham bone, and chicken bouillon; simmer until meat falls off the bone, about 1 hour. Transfer ham bone to a cutting board and allow to cool slightly.
Stir beans and potatoes into ham broth and simmer for 10 minutes. Meanwhile, cut ham into bite-size pieces and return to the pot. Add tortellini to the ham and bean soup; simmer until pasta is tender, 7 to 9 minutes. Stir zucchini and parsley into soup and continue to simmer until zucchini is tender, about 5 minutes more.
Divide soup into bowls and garnish with Parmesan cheese and black pepper.