Bring a large pot of salted water to a boil and cook the broken spaghetti according to package directions until al dente, about the time indicated on the package.
Drain the pasta in a colander and rinse well under cold water to stop cooking and cool the noodles; shake off excess water.
Place the cooled spaghetti in a large mixing bowl.
Add the sliced red onion, sliced cucumbers, chopped red, green, and yellow bell peppers, halved cherry tomatoes, sliced black olives, sweet corn, and chopped parsley to the bowl.
Pour the Italian salad dressing over the pasta and vegetables and toss gently until everything is evenly coated; add more dressing to taste.
Sprinkle the freshly grated Parmesan cheese on top and toss lightly or leave it on top as a garnish.
Serve immediately or refrigerate until ready to serve.