While you're gathering everything, bring a large pot of salted water to a boil and add the pasta, cooking it until just al dente according to package directions. Before draining, reserve 1 cup of the starchy pasta water—this is crucial for adjusting the sauce consistency later. While the pasta cooks, slice the squash into 1/4-inch rounds, dice the onion, and mince the garlic cloves. Freshly minced garlic makes a noticeable difference in flavor compared to pre-minced, so it's worth the extra minute.
Heat the olive oil in a large pan over medium-high heat and add the diced onion, cooking for about 1 minute until just starting to soften. Add the sliced squash, minced garlic, salt, black pepper, and crushed red pepper flakes, then cook for 15-20 minutes, stirring occasionally, until the squash is completely tender and deeply browned on the edges. The browning develops deeper flavor, and the squash will break down significantly, which is exactly what you want for a smooth sauce.
Transfer the cooked squash mixture to a blender (working in batches if needed) and blend until completely smooth, creating a velvety sauce base. I like to leave it slightly warm when blending rather than letting it cool completely, as it blends more easily and the warmth helps the cheese melt evenly in the next step.
Pour the blended squash back into the large pan over low heat and stir in the grated Parmesan cheese, mixing constantly until the cheese melts completely and the sauce becomes creamy and cohesive. I prefer grating Parmesan from a block rather than using pre-shredded, as it melts more smoothly without any cellulose additives. Add the cooked pasta from Step 1 and fold it gently into the sauce. If the mixture seems too thick, add reserved pasta water 1/4 cup at a time, stirring well between each addition, until you reach a silky, coating consistency.
Remove the pan from heat and stir in the lemon juice, which brightens the squash flavor and cuts through the richness of the cheese. Taste and adjust seasoning with additional salt and pepper as needed. Fold in the fresh chopped parsley just before serving to maintain its vibrant color and fresh flavor.