Heat olive oil in a large pot over medium heat. Add the onion and garlic, sautéing until fragrant and translucent (about 5 minutes).
Add the mushrooms and cook until they release their moisture and begin to brown (about 10 minutes).
Add the diced potatoes, vegetable broth, and thyme. Bring to a boil, then reduce heat and simmer for 20 minutes, until the potatoes are tender.
Use an immersion blender to puree the soup directly in the pot until smooth, or transfer to a blender in batches.
Stir in the coconut milk or cream for extra creaminess, and season with salt and pepper.
Serve hot, garnished with fresh parsley if desired.