Heat oil in a large Dutch oven over medium-high heat. Add mushrooms, pepper and salt; cook, stirring occasionally, until tender and golden brown, about 8 minutes. Add vinegar; cook, stirring to scrape up any browned bits from the bottom of the pan, for about 30 seconds. Stir in broth; bring to a boil over medium-high heat. Add penne and chicken; cover and cook, stirring occasionally, until the pasta is al dente, 10 to 12 minutes. Remove from heat.
Stir in crème fraîche (or sour cream) and Parmesan. Divide the pasta evenly among 4 bowls; garnish with parsley, if desired.
