Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs for al dente. Ladle 1 1/2 cups cooking water into a liquid measuring cup and set aside, then drain the pasta in a colander.
Meanwhile, heat the olive oil and garlic together in a large pot or Dutch oven over medium-high heat. Cook, stirring, until the garlic is browned, 1 to 2 minutes. Add the squash and 1/2 teaspoon salt. Reduce the heat to medium, cover and cook, stirring occasionally, until the squash is very tender and starting to break down, 6 to 8 minutes.
Add 3/4 cup of the reserved cooking water to the squash and bring to a simmer. Cook, stirring and breaking up the squash with a wooden spoon, until the sauce is slightly thickened, about 1 minute. Add the pasta and toss until well coated. Add more cooking water to loosen, as needed. Season with salt.
Divide the pasta among bowls. Drizzle with more olive oil and sprinkle with the Parmesan.
Timing is important when you’re making pasta. Start cooking the garlic and squash as soon as you add the pasta to the boiling water so that everything is ready at once.