Heat the olive oil in a large skillet over medium-high heat. Add the sausage and cook, breaking up any clumps with a wooden spoon, until brown, about 5 minutes; transfer to a bowl with a slotted spoon. Add the bell peppers to the skillet with 1/4 teaspoon salt and cook, stirring occasionally, until tender, about 4 minutes. Set aside.
Whisk together the half-and-half, Parmesan, parsley, eggs, 1 1/2 teaspoons salt and a few grinds of pepper in a medium bowl.
Layer half of the bread cubes in the prepared baking dish. Sprinkle with 1 1/2 cups of the mozzarella, followed by the sausage and peppers. Cover with the remaining bread cubes and pour the custard evenly over the top. Cover with plastic wrap and let soak at room temperature 30 minutes to 1 hour. (To make ahead, after adding the custard, cover with plastic wrap and refrigerate overnight and then let the strata sit at room temperature while the oven preheats.)
Preheat the oven to 325 degrees F. Remove the plastic and sprinkle with the remaining 1/2 cup mozzarella. Bake until slightly puffed and no longer jiggly in the center, 40 to 50 minutes. Let stand 5 minutes before serving.
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