1. Cook pasta according to package directions. Reserve 1/2 cup cooking liquid; drain.
2. Heat the olive oil in a large nonstick skillet over medium heat. Add onion and saffron; cook until onions are softened (about 5 minutes), stirring frequently. Add garlic; cook until fragrant (about 30 seconds), stirring constantly. Stir in bell peppers, capers, vinegar, thyme, salt, and pepper; cook until heated through (about 3 minutes), stirring occasionally.
3. Add the pasta and the reserved pasta liquid to sauce in the skillet; cook until heated through (about 2 minutes), stirring constantly. Transfer mixture to serving bowls; top with Parmesan curls. Serve immediately.