The stock for this recipe comes from saved bean water from home cooked dry beans. You can substitute any stock or broth you like, but water will not provide enough flavor on its own. (It probably goes without saying that you can also use canned beans in this recipe, but be sure to rinse them well before adding them to the soup.) I've learned to leave the pasta out of the minestrone and to portion it into the serving bowls instead. That way the pasta doesn't swell and drink up all the broth.
Heat the olive oil in a saucepan over medium-high heat. Cook the onion, carrots and garlic, stirring for about 4 minutes until the onion turns translucent but the vegetables do not brown.
Bring the bean water and tomatoes with their juice to a simmer. Add the parmesan rind, if using, salt and pepper and check for seasoning, adding additional salt to suit your taste. (You can prepare the soup in advance up to this point.) Simmer for about 15 minutes.
Add your choice of green vegetable and simmer until cooked through but the vegetables still have some life and texture, about 15 minutes.
To serve, portion the pasta into serving bowls and ladle the hot soup over the pasta. Top with a portion of parmesan cheese, if using.