Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs for al dente. Reserve 1 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large Dutch oven over medium-high heat. Add the garlic and sage and cook until the garlic is light golden and the sage is wilted, about 2 minutes. Add the onion, celery and carrot and season with salt and pepper. Cook, stirring occasionally, until almost tender, about 6 minutes. Add the pork, beans with their liquid and the chicken broth; bring to a boil. Reduce to a gentle simmer and cook until the vegetables are tender, about 5 minutes.
Add the pasta and 1/2 cup cooking water to the bean mixture. Stir to combine, adding more cooking water as needed to thin the broth. Season with salt and stir in the parsley. Divide among bowls, drizzle with olive oil and season with pepper.