Heat 2 tablespoons of butter in a large skillet over medium heat. Add the onion, carrots, and garlic. Cook until the onion softens.
Add the mushrooms and cook until they release their liquid and start to brown. Stir in the flour and tomato paste until everything is coated.
Pour in the red wine and vegetable broth. Add the thyme and bay leaf. Season with salt and pepper. Bring to a simmer and cook for 20 minutes until the sauce thickens.
While the sauce simmers, cook the egg noodles in a pot of salted water according to package directions. Drain and return them to the pot.
Toss the hot noodles with the remaining 2 tablespoons of butter and the Parmesan cheese. Divide the noodles among four bowls and spoon the mushroom mixture over the top. Sprinkle with parsley before serving.