Cook the pasta according to the package directions; drain and return it to the pot.
Heat the canola oil in a large nonstick skillet over medium-high heat. Cook the chicken until browned, 4 to 6 minutes; transfer to a plate.
Add the olive oil, onion, and mushrooms to the skillet; season with ¼ teaspoon each salt and pepper. Cook, tossing occasionally, until tender, 4 to 6 minutes.
Return the chicken to the skillet, add the tomatoes and broth, and cook until the sauce has thickened, 3 to 4 minutes.
Add the chicken and vegetables, Parmesan, and parsley to the pasta and toss to combine.