Working one at a time, place chicken on a cutting board; slice each in half horizontally to create four thin cutlets. Sprinkle chicken evenly on both sides with 1½ teaspoons salt and ½ teaspoon pepper. Place in a shallow dish with flour; toss and pat to coat chicken with flour. Heat ¼ cup olive oil over medium-high heat until shimmering. Add chicken to pan; cook until browned on both sides, 3 to 4 minutes per side. Place chicken on a wire rack to drain. Wipe skillet clean, if needed. Reduce heat to medium. Add remaining 2 tablespoons oil and butter. Add mushrooms; sprinkle with remaining ½ teaspoon each salt and pepper. Cook, stirring occasionally, until mushrooms are lightly browned, about 5 minutes. Add garlic; cook, stirring often, about 1 minute. Add flour; cook, stirring often, until flour is slightly toasted, about 1 minute. Add marsala to pan; stir to release browned bits from bottom of pan. Add chicken stock. Reduce heat to medium-low; simmer until slightly thickened, about 10 minutes. Stir in cream; return chicken to pan. Cook until chicken is warmed through, about 2 minutes. Remove from heat. Divide chicken and sauce among 4 plates. Sprinkle with parsley, and serve.