Cook pasta according to package directions until al dente. Reserve 1/2 cup of pasta water and drain the rest.
In a large skillet, heat olive oil over medium heat. Add onion and cook until soft, about 5 minutes.
Add garlic and mushrooms. Cook until mushrooms release their juices and start to brown, 7–8 minutes.
Sprinkle in thyme and chili flakes. Stir well.
Pour in vegetable broth and let it reduce for about 5 minutes.
Stir in coconut cream and let the sauce simmer until slightly thickened, about 5 minutes.
Season with salt and black pepper. Toss in the cooked pasta, adding reserved pasta water as needed for creaminess.
Serve warm, topped with parsley and parmesan.