Tear off and reserve the outer leaves from the Brussels sprouts, then chop the rest. Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs, adding the chopped Brussels sprouts and their leaves during the last 2 minutes of cooking. Reserve 11/2 cups cooking water, then drain.
Meanwhile, heat 1 tablespoon olive oil in a large skillet over medium heat. Add the bacon and cook, stirring occasionally, until crisp, about 5 minutes. Remove to a paper towel¿lined plate, reserving 1 tablespoon drippings in the skillet.
Add 1 tablespoon olive oil to the skillet, then add the squash, garlic and 1/2 teaspoon salt. Cook until the squash is lightly golden around the edges, about 1 minute. Add 1 cup of the reserved cooking water, cover and simmer until the squash is tender, 4 to 5 minutes.
Add the pasta, Brussels sprouts, bacon, scallions, the remaining 1 tablespoon olive oil and 1/4 cup cooking water to the skillet; toss until coated, adding the remaining cooking water as needed to loosen. Season with salt and pepper. Top each serving with the parmesan.