Cut yellow squash and zucchini in half lengthwise and then into half moon shapes. Cut red bell pepper and red onion into approximately 2-inch chunks. Place together in a medium bowl and toss with 1 tablespoon olive oil. Sprinkle with salt and pepper.
Heat a grill pan over medium-high heat. When hot add vegetables. Grill until they develop nice grill marks and then flip over and continue to cook until softened. Place in a large bowl.
While vegetables are grilling, bring a large pot of water with at least 1 tablespoon of salt to a boil. Cook pasta to al dente. Drain pasta and rinse with water. Drain well and add to bowl with grilled vegetables.
In a medium bowl, combine lemon juice, red wine vinegar, Dijon mustard, sugar, garlic powder, salt, pepper, and thyme. Gradually whisk in oil. Add basil.
Pour dressing over pasta and vegetables. Add tomatoes and Parmesan cheese. Refrigerate for at least 1 hour.