Rinse and pat the sole fillets dry with paper towels and sprinkle one side with salt.
Combine gf flour, 2 teaspoons of salt, and 1 teaspoon of pepper in a shallow bowl. Dredge sole fillets in the seasoned flour.
Heat 3 tablespoons of butter in a large (12-inch) skillet over medium heat until it starts to brown. Add 2 floured fillets and cook until golden on bottom, 2 to 3 minutes. Carefully turn fish over and cook until opaque in center and golden on bottom, about 2 more minutes. While the second side cooks, add 3 tablespoons of lemon juice to the pan.
Carefully transfer fish fillets to a plate and pour sauce from the skillet over the fish. Sprinkle with parsley, salt, and pepper.
Keep warm while you repeat steps 3 and 4 with the remaining 2 floured fillets.
For extra sauce, add 1/4 cup of lemon juice to the skillet, stir, scraping up the little bits. Serve sauce on the side.