In a large pot of boiling, salted water, cook the pasta for 7 minutes (the pasta will only be partially cooked). Drain, reserving 2 1/2 cups of the pasta cooking water.
Meanwhile, in an extra-large skillet, cook the bacon over medium-high heat until crisp, about 5 minutes. Drain on paper towels, then crumble the bacon once it is cool enough to handle. Discard all but 3 tablespoons of the bacon fat in the pan.
Add the olive oil and squash to the skillet. Cook, stirring occasionally, until browned, about 7 minutes. Add the pasta and 2 cups of the reserved pasta cooking water. Bring to a boil and cook, stirring, until the pasta and squash are cooked, about 5 minutes, adding additional pasta cooking water if needed.
Remove from the heat and stir in the spinach, garlic and parsley. Season generously with salt and pepper; sprinkle with the cheese and crumbled bacon.