Bring a pot of salted water to a boil. Add the ravioli and cook as the label directs; drain, then drizzle with olive oil and toss.
Meanwhile, season the chicken with salt and pepper. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the chicken; cook, undisturbed, until beginning to brown, about 2 minutes. Continue to cook, stirring, 1 more minute. Transfer to a plate.
Heat the remaining 1 tablespoon olive oil in the skillet. Add the mushrooms and cook, undisturbed, until browned in spots, about 2 minutes. Season with salt and continue to cook, stirring, until softened, about 3 more minutes. Stir in the tomatoes, garlic and vinegar and cook until the tomatoes begin to soften, about 2 minutes. Return the chicken to the skillet, then add the ravioli, broth and Parmesan; bring to a simmer and cook, stirring occasionally, until the chicken is cooked through, about 4 minutes. Top with the parsley.