1. Heat 2 tablespoons of the olive oil in a large saucepan. Add the onion and garlic and cook over moderate heat, stirring frequently, until lightly browned, about 5 minutes. Add the tomatoes, stock, water and cranberry beans and bring to a boil. Cover partially and cook over moderately low heat until the cranberry beans are tender, about 15 minutes. Season with salt and pepper. Add the string beans and flat beans, cover partially and cook over moderate heat until tender, 10 to 12 minutes.
2. Meanwhile, bring a medium saucepan of salted water to a boil. Add the pasta and cook, stirring, until al dente. Drain the pasta and add it to the beans along with the parsley, vinegar, 1/2 cup of Parmesan and the remaining 3 tablespoons of olive oil. Season with salt and pepper. Serve the minestra in deep bowls and pass extra Parmesan at the table.
Make Ahead: The minestra can be prepared through Step 1 and refrigerated for up to 2 days.
Wine Recommendation: Look for a Chianti with some spice and good acidity to stand up to the tomatoes and vinegar in the soup. Two choices: the 1997 Castellare di Castellina Chianti Classico or the 1997 Badia a Coltibuono Chianti Classico.