Preheat oven to 400°F. Line a baking sheet with parchment paper.
In a medium sized bowl, toss the diced squash with 2 tablespoons of the olive oil.
Place the squash on the prepared baking sheet in a single layer and season with salt and pepper. Roast, flipping half way through for 30-35 minutes or until the squash is tender and caramelized. Set aside.
Meanwhile, warm the broth in a small saucepan over low heat.
In a large saucepan, heat the remaining tablespoon of olive oil over medium heat. Add the onion and garlic and cook until softened and fragrant, about 5 minutes. Add the thyme, stirring and cooking until fragrant, about 1 minute.
Add the barley to the pan and cook for about 2 minutes or until lightly toasted, stirring often. Add the wine and cook until it is evaporated, stirring occasionally.
Reduce the heat to medium-low and add 1 cup of the broth, stirring frequently, until the liquid is completely absorbed, about 5 minutes. Continue adding the rest of the broth, 1 cup at a time, allowing each addition of the broth to be absorbed before adding more.
Remove from the heat when the barley is tender and the risotto is creamy. Stir in the cooked squash and grated parmesan cheese. Season with salt and pepper to taste. Garnish with finely chopped parsley and additional parmesan cheese if desired. Serve immediately.