Bring 4 quarts of water and 1 tablespoon salt to a boil. Add pasta and 1 minute pasta al dente, about 8 minutes. Drain well, rinse with cold water, and drain again.
Meanwhile, combine broccoli florets, bell peppers, and onion in a large bowl. Stir in cold pasta. Add Parmesan cheese and salad dressing and toss until uniformly combined.
Place in a serving bowl and cover with plastic wrap. Chill at least 2 hours before serving.
Stir carefully before serving. Add more salad dressing as needed to keep the pasta from drying out.
To blanch broccoli, bring 2 quarts water and 1 tablespoon salt to boil. Add broccoli and cook until just tender and bright green, about 3 to 4 minutes. Drain and immediately plunge into an ice bath to stop the cooking.