Preheat the oven to 425 degrees F.
Pulse the bread cubes in a food processor to make coarse crumbs (you should have about 1/2 cup). Set aside.
Remove and finely chop the mushroom stems (you should have about 1 cup); set aside. Wipe the mushroom caps clean with a damp paper towel. With a paring knife, cut a thin slice from the top of the mushroom caps so they'll sit flat when stuffed. Put the mushroom caps in a large bowl, drizzle with 2 tablespoons of the olive oil and 1/2 teaspoon salt, and toss to coat in the oil.
Heat the remaining 2 tablespoons olive oil in a medium skillet over medium heat. Add the sausage and cook, crumbling with a wooden spoon, until no longer pink, about 3 minutes. Add the onions, peppers, garlic and chopped mushroom stems and cook until the vegetables are tender, 6 to 7 minutes. Scrape into a bowl and let cool.
Once the sausage mixture has cooled, stir in the breadcrumbs, mozzarella, 2 tablespoons of the Parmesan, the parsley, 1/4 teaspoon salt and some pepper. Mix well. Divide the stuffing among the mushroom caps. Arrange the stuffed caps in an oiled 9-by-13-inch baking dish and sprinkle with the remaining 2 tablespoons Parmesan. Bake until the mushrooms are tender and the filling is browned, 20 to 25 minutes. Serve warm or at room temperature.