Preheat oven to 400° F. Coat a rimmed baking sheet with cooking spray.
Place potatoes and broccoli on the baking pan and toss with 1 tablespoon oil and 1/4 teaspoon salt. Cook, stirring once, until tender, 20 to 25 minutes.
Meanwhile, pat cod dry and season with the remaining 1/4 teaspoon salt and pepper. Add cod and tomatoes to baking pan and drizzle with 1 tablespoon of oil. Bake for 10 minutes per inch thickness or until fish is done and flakes easily.
While fish is cooking, make the lemon vinaigrette by combining the remaining 2 tablespoons oil, lemon juice, mustard, garlic, and thyme in a small bowl.
Serve the vegetables and cod drizzled with the lemon vinaigrette and garnished with Parmesan.