Heat 1 teaspoon vegetable oil in a large skillet over medium heat; cook and stir carrots 2 to 3 minutes. Add zucchini and yellow squash; cook until tender, 5 to 10 minutes.
Heat chicken broth in a large saucepan over medium heat; reduce heat to low and keep warm.
Heat 1 teaspoon vegetable oil in a separate large skillet over medium heat; cook and stir onion until lightly browned and tender, about 10 minutes. Add garlic and thyme; cook until fragrant, 1 to 2 minutes. Cook and stir barley into onion mixture until lightly toasted, 2 to 3 minutes.
Pour about 2 cups warm broth into barley mixture; reduce heat to low, and simmer, stirring occasionally, until broth is absorbed, 5 to 10 minutes. Add remaining broth, 1/2 cup at a time, allowing broth to absorb before each addition, until barley is tender, 15 to 20 minutes. Stir in carrot mixture, butter, and Parmesan cheese until cheese melts, 1 to 2 minutes. Season with salt and black pepper.
