Whisk lemon juice and lemon zest together in a bowl. Mix flour, salt, and pepper together in a resealable gallon-size plastic bag.
Pat swai fish fillets dry with a paper towel; lightly season 1 side of each fillet with salt. Place fillets in the bag with flour mixture, 1 at a time, and shake to coat.
Heat a large skillet over medium heat; add butter and heat until lightly browned. Place coated fish fillets in the browned butter; cook until lightly browned on the edges and fish flakes easily with a fork, 2 to 3 minutes per side. Transfer fish to a plate, reserving butter in the skillet.
Pour lemon juice mixture into skillet with reserved butter; cook and stir until a light sauce forms, 2 to 3 minutes. Spoon sauce over fish and garnish with parsley.