Season the pork chops with salt and pepper; sprinkle all over with 1 1/2 tablespoons flour. Heat 1/2 tablespoon olive oil in a large nonstick skillet over medium-high heat. Working in 2 batches, add the chops and cook until golden and just cooked through, 3 to 4 minutes per side, adding 1/2 tablespoon olive oil to the skillet between batches. Transfer the chops to a plate and keep warm.
Heat the remaining 1/2 tablespoon olive oil in the same skillet; add the mushrooms, garlic, scallion whites and a pinch of salt and cook, without stirring, until the mushrooms begin to brown, about 3 minutes. Sprinkle in the remaining 1/2 tablespoon flour and cook, stirring occasionally, until the mushrooms are soft, about 3 more minutes. Stir in the chicken broth and sour cream, bring to a simmer and cook until the gravy thickens, about 1 more minute.
Divide the pork chops among plates. Drizzle any juices from the pork into the gravy; stir in the scallion greens and parsley. Season with salt and pepper; spoon over the pork. Serve with cornbread.