In a small bowl, soak mushrooms in 1 cup hot water until soft, about 15 minutes. Gently rub under water to release grit, then lift out (reserve liquid); finely chop mushrooms.
Meanwhile, pour oil into a 6- to 8-quart pan over medium-high heat. When hot, add veal, pork, parsley, and garlic. Stir constantly, breaking meat into small pieces, until browned, 5 to 8 minutes.
Stir in tomatoes, oregano, 1/4 teaspoon salt, 1/8 teaspoon pepper, and mushrooms; carefully pour in reserved soaking liquid, leaving grit behind. Bring to a boil, then lower heat, cover, and simmer, stirring occasionally, until tomatoes are broken down and sauce is thick, about 1 1/2 hours. Add salt and pepper to taste.