Poor Man's Pasta Fagioli

5 ½ cups water

1 pound ditalini pasta

4 (4 ounce) cans tomato sauce (such as Hunt's®)

1 (15 ounce) can cannellini beans

1 (15 ounce) can kidney beans

1 (15 ounce) can black-eyed peas

1 ½ cups frozen green peas

1 (15.25 ounce) can whole kernel corn

1 large potato, diced

2 carrots, diced

1 onion, diced

1 small head of garlic, peeled and minced

1 teaspoon chopped fresh parsley, or to taste

1 pinch Italian seasoning

1 pinch salt and ground black pepper

1 teaspoon extra-virgin olive oil

1 teaspoon grated Parmesan cheese, or to taste