Sprinkle chicken with ⅛ teaspoon each salt and pepper. Heat sun-dried tomato oil in a large skillet over medium heat. Add the chicken and cook, turning once, until browned and an instant-read thermometer inserted into the thickest part registers 165°F, about 6 minutes total. Transfer to a plate.

Add sun-dried tomatoes and shallots to the pan. Cook, stirring, for 1 minute. Increase heat to high and add wine. Cook, scraping up any browned bits, until the liquid has mostly evaporated, about 2 minutes. Reduce heat to medium and stir in cream, any accumulated juices from the chicken and the remaining ⅛ teaspoon each salt and pepper; simmer for 2 minutes. Return the chicken to the pan and turn to coat with the sauce. Serve the chicken topped with the sauce and parsley.