Cook the pasta: Bring a large pot of salted water to a boil. Add the spaghetti and cook until al dente according to package directions.
Drain well.
Cool the pasta: Rinse the spaghetti briefly under cold water to stop the cooking and cool it down. Shake off excess water. Toss with a splash of olive oil to prevent sticking.
Make the dressing: In a jar or bowl, whisk together olive oil, red wine vinegar, lemon juice, garlic, Dijon, honey, oregano, basil, red pepper flakes, salt, and pepper until emulsified.
Prep the veggies: Halve the tomatoes, dice the cucumber and bell peppers, and thinly slice or finely dice the red onion.
Slice the olives.
Toss it all together: In a large bowl, combine the cooled spaghetti, tomatoes, cucumber, peppers, onion, olives, and parsley. Pour over most of the dressing and toss until coated.
Add cheese and adjust: Sprinkle in the Parmesan and toss again. Taste and add more dressing, salt, or pepper as needed.
Chill: Cover and refrigerate for at least 1 hour, preferably 2–4 hours.
The flavors meld and the pasta absorbs the dressing.
Serve: Before serving, toss again. If it seems dry, splash in a bit more dressing or a drizzle of olive oil and a squeeze of lemon.