Bring a large pot of water to a boil. Season water generously with salt. Add ravioli and cook according to package directions. Drain, drizzle with 2 to 3 tablespoons vegetable oil, toss, and set aside.

Meanwhile, add 1 tablespoon vegetable oil to a large skillet over medium heat. Season chicken with salt and pepper; add to skillet. Cook chicken until all sides are brown and no longer pink in the middle. Transfer chicken to a plate; set aside.

Heat 1 tablespoon oil in same skillet. Add asparagus and cook for 4 minutes turning a couple times. Add sun-dried tomatoes, garlic, and vinegar; cook for 2 minutes. Add the chicken back to the skillet, along with the drained ravioli, broth, and Parmesan cheese. Bring to a simmer, and cook for 4 minutes, stirring occasionally. Sprinkle with parsley; serve warm.