Preheat the oven to 400°F and lightly oil a baking dish.
Cut the chorizo into small pieces and put them in a food processor. Process until the chorizo is chopped, then set aside.
Remove the stems from the mushrooms. Place half of the stems in the food processor and process. Discard the other half of the stems. Brush the mushrooms with olive oil and place on the baking dish and set aside.
Place the olive oil into a medium pan over medium-high heat. Add the onion and cook, stirring for about 3 minutes. Add the chorizo and mushroom stems.
Season with salt and pepper and cook for about 3 minutes more. Add the bread crumbs and stir. Cook for about 1 minute, add the heavy cream and mix well. Place the mixture in a bowl and set aside.
Add the parsley and cheese. Using a small spoon fill the mushroom. Bake, uncovered until stuffing is golden, 25 to 30 minutes. Garnish with scallions and serve hot or warm.