Heat the oven to 450°F and arrange a rack in the middle. Place a baking sheet upside down in the oven while it’s heating.
Remove chorizo from its casing and place in a medium frying pan over medium-high heat. Cook, stirring occasionally, until browned, about 7 minutes. Remove to a plate and set aside.
Add 2 tablespoons of the olive oil to the pan. Add mushrooms and season with salt and freshly ground black pepper. Cook, stirring rarely, until beginning to brown, about 10 minutes.
Roll out pizza dough to a 10-inch round, remove the hot baking sheet from the oven, drizzle with remaining 1 tablespoon olive oil, and carefully place dough on the upside-down baking sheet. Scatter chorizo, mushrooms, Jack cheese, and Parmesan cheese on top. Drizzle with olive oil and bake until cheese is melted and crust is crisp and browned, about 15 minutes.