Preheat the oven to 400 degrees. Toss the turnip and squash cubes with 2 tablespoons of the olive oil, the balsamic vinegar, thyme, salt, and pepper on a baking sheet.
Roast the vegetables for 25 to 30 minutes until they soften and the edges begin to brown.
While the vegetables roast, heat the remaining tablespoon of olive oil in a large pot over medium heat. Add the onion and cook for 5 minutes until it softens.
Stir in the garlic and cook for 1 minute more. Pour in the vegetable broth and bring the mixture to a simmer.
Add the roasted turnip and squash to the pot. Simmer everything together for 10 minutes.
Use an immersion blender to puree the soup until smooth. Adjust salt and pepper if needed.
Ladle the soup into bowls and top each serving with chopped fresh parsley.