Place each chicken breast between 2 sheets of parchment paper and pound to an even 1/2-inch thickness. Sprinkle with salt. Coat chicken on both sides with flour; shake off excess.
Warm 2 Tbsp. oil in a large skillet over medium-high heat. Cook chicken until golden on both sides and cooked through, 8 to 10 minutes. Transfer to a plate; cover.
Heat remaining oil in skillet. Sauté garlic until fragrant, 30 seconds. Add peppers and sauté for 1 minute. Add broth, vinegar and sugar; bring to a boil, scraping up browned bits from pan. Reduce heat; simmer until slightly thickened, about 4 minutes. Remove pan from heat; stir in butter and parsley until butter melts. Pour sauce over chicken and serve.