Heat olive oil in a large pot over medium heat.
Sauté onion until translucent, about 3-4 minutes. Add garlic and cook for 1 minute.
Stir in mushrooms and cook until golden brown and softened, about 5-7 minutes.
Add vegetable broth and bring to a simmer. Cook for 10 minutes.
Stir in almond milk and thyme. Blend part of the soup with an immersion blender if desired.
For a thicker consistency, dissolve cornstarch in 1 tbsp water, then stir into the soup and simmer for 2 minutes.
Season with salt and pepper, garnish with parsley, and serve warm.