Bring a large pot of lightly salted water to a boil. Add the pasta and cook until just tender, about 8 to 10 minutes.
While the pasta cooks, place the oil in a very large skillet set over medium heat. Add the mushrooms, garlic and oregano and cook until the mushrooms are tender and no moisture is left in the pan. Add the wine and reduce to about 2 Tbsp. Add the cream and bring to a simmer.
When the pasta is cooked, reserve 1/2 cup of its cooking liquid, and then drain the pasta well. Add the pasta and reserved cooking liquid to the skillet, season with salt and pepper, and toss to combine. Spoon the pasta into bowls, and top with nuggets of cheese, pine nuts and parsley. If desired, at the table, let diners top the pasta with Parmesan cheese.
Note: The stems of shitake mushrooms are tough and should be removed and discarded. Trim off and discard the tough, lower parts of the oyster mushroom stems.
Note: To toast the pine nuts, place in a single layer in a baking pan. Bake, shaking the pan from time to time, in a 350 degrees F oven for 10 to 15 minutes, or until lightly toasted.