Place chicken between 2 sheets of waxed paper and pound until thin. Sprinkle chicken with salt and pepper and dredge in flour.
Heat 1 Tbsp. butter and 1 Tbsp. vegetable oil in a large skillet over medium-high heat until butter foams. Add 2 chicken breast halves and cook without moving until browned, 3 minutes. Turn and cook until firm and browned on both sides, 3 minutes more. Transfer to a plate and cover loosely with foil to keep warm.
Add 1 Tbsp. butter and remaining 1 Tbsp. oil to skillet; cook remaining chicken breast halves. Transfer to plate and cover to keep warm.
Add lemon juice and broth to skillet and bring to a boil, scraping up any browned bits from bottom of pan with a wooden spoon. Boil, stirring occasionally, until thickened, about 5 minutes. Remove from heat, add remaining 1 Tbsp. butter and parsley and stir until butter melts. Season with salt and pepper. Pour sauce over chicken and serve immediately.