Preheat the oven to 350 degrees F (175 degrees C).
Cut each roll into 8 cubes. Spread onto a baking sheet in a single layer.
Bake in the preheated oven until toasted, about 10 minutes.
Layer toasted bread cubes in a 1 1/2-quart baking pan. Sprinkle 1/4 cup Swiss cheese, 1/4 cup Cheddar cheese, 1/2 the bacon, and 2 green onions on top of the bread. Shake to settle the toppings into the bread.
Beat eggs together with half-and-half, milk, salt, and pepper in a bowl. Pour over the bread cubes, pressing cubes down into the mixture. Sprinkle the remaining Swiss cheese, Cheddar cheese, bacon, green onions, and Parmesan cheese on top.
Bake in the preheated oven until strata is hot in the center, about 30 minutes.
You can toast the bread up to 24 hours in advance.
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