Bring a large stockpot of salted water to a boil and cook the broken spaghetti according to package directions until al dente.
Drain the pasta, rinse under cold water to stop cooking, shake or pat dry to remove excess water, and set aside.
While the pasta cooks, prepare the vegetables: slice and quarter the cucumber, halve the cherry tomatoes, dice the red and green bell peppers, finely dice the red onion, and chop the parsley.
In a very large mixing bowl, combine the drained olives, cucumber, cherry tomatoes, red bell pepper, green bell pepper, red onion, crumbled feta, grated Parmesan, and parsley.
Add the cooled, drained spaghetti to the bowl with the vegetables and toss gently to combine.
Pour the Italian dressing over the salad and toss again until everything is evenly coated.
Taste and season with salt and freshly ground black pepper; add lemon juice and/or balsamic vinegar if desired and toss to combine.
Cover the bowl and refrigerate for at least 45 minutes (up to overnight) to chill and allow flavors to meld before serving.
