Combine flour and ¼ teaspoon salt in a shallow bowl.
Dredge flounder fillets in the seasoned flour until well coated on both sides, shaking off any excess flour.
Add olive oil and ½ tablespoon butter to a large skillet over medium-high heat.
Once oil is hot, add flounder fillets to the pan and cook for 3-4 minutes per side, until golden. (Depending on the size of your pan, you may need to do this in batches.) Transfer fish to a plate and set aside.
Reduce heat to medium and add wine, lemon juice, capers, and remaining ⅛ teaspoon salt to the same pan. Cook for about 2 minutes.
Stir in the remaining ½ tablespoon butter until melted.
Return flounder fillets to the pan and let cook in the sauce for 2 more minutes, turning halfway through.
Plate flounder, drizzle on the remaining pan sauce, and top with parsley.