Combine the flour, grated Parmesan, garlic powder, 1/2 teaspoon salt, and 1/2 teaspoon pepper in a shallow bowl.
Cut each chicken breast into 3 pieces. Place a piece between two sheets of plastic wrap and pound to about 1/4-inch thickness; repeat with all pieces.
Rub the chicken pieces with 1 tablespoon olive oil then dredge evenly in the flour-Parmesan mixture, shaking off excess.
Heat 2 tablespoons olive oil and 1 tablespoon butter in a large skillet over medium-high heat until shimmering.
Add half the coated chicken to the skillet and cook 3–4 minutes on the first side until golden brown, then flip and cook 1–2 minutes more until cooked through. Transfer to a plate, season lightly with salt if desired, and cover with foil. Repeat with the remaining chicken, adding more oil or butter if needed.
Pour off excess fat from the skillet and wipe lightly with paper towels. Add 1 teaspoon olive oil and the minced garlic and sauté about 20 seconds until fragrant.
Pour in the chicken stock, lemon juice, and lemon zest, then add the drained capers. Bring to a simmer and cook 3–4 minutes until slightly reduced.
Reduce heat to low, add 2 tablespoons butter, and swirl the pan until the butter melts and the sauce becomes glossy and slightly thickened. Stir in the parsley if using.
Arrange the cooked chicken on plates, spoon the lemon-caper sauce over the top, and serve immediately.