If using fresh peas, parboil them in salted water for 3 to 5 minutes. Drain them and set aside. Trim the tough ends and wilted spots from the mushrooms. Wipe them clean, or wash them quickly and air dry them. Slice thinly and set aside.

Bring 6 quarts of salted water to a boil.

Heat olive oil in a large skillet over medium heat. Crush the garlic cloves and add them along with the prosciutto in the skillet, and cook, stirring, until lightly browned, about 4 minutes. Stir in the mushrooms, season with salt and pepper, and cook until wilted, about 7 minutes.

Add the tomatoes, season, and bring to a boil. Lower the heat to medium, and cook for 5 minutes. Stir in the peas and chopped parsley, and cook until peas are tender, about 3 minutes.

While the sauce is simmering, cook the bucatini in boiling water. Stirring frequently, cook pasta for 10 minutes.

To serve, drain pasta, and return to pot. Add approximately 3/4 of the sauce to the pasta and bring to a boil. Season pasta with salt and pepper. Remove pot from heat and add the grated Parmesan to pasta and stir. Transfer pasta to a large platter, top with remaining sauce, and serve immediately.