Heat olive oil in a large pot over medium heat. Add the sliced onions and cook, stirring frequently, until they are caramelized, about 20 minutes.

Add the minced garlic, diced carrots, and diced celery to the pot. Cook for an additional 5 minutes until the vegetables are softened.

Stir in the sliced mushrooms and thyme. Cook until the mushrooms are tender and have released their moisture, about 10 minutes.

Pour in the vegetable broth and balsamic vinegar. Bring to a boil, then reduce heat and let simmer for 15 minutes.

Using a blender, carefully blend the soup until smooth. Return to the pot and stir in the heavy cream. Heat through without boiling.

Season with salt and pepper to taste. Serve hot, garnished with chopped parsley.