Cut one lemon in half and squeeze the juice into a small bowl. Slice the other lemon into thin rounds and set the slices and the lemon juice aside.
Rinse the mahi-mahi fillets with cold water and pat them dry with paper towels.
Season both sides of each fillet with salt and ground black pepper.
Heat 2 tablespoons olive oil in a skillet or nonstick frying pan over medium heat.
Add the fillets to the skillet and cook undisturbed for 5 minutes. Flip the fillets gently and cook for another 5 minutes, until both sides are golden brown and the fish is cooked through.
Transfer the cooked fillets to a plate and let them rest while you make the sauce.
Reduce the heat to medium-low. Add the reserved lemon juice, 2 cloves minced garlic, and a pinch of salt to the same skillet. Stir constantly until the garlic becomes aromatic (about 30 seconds).
Add the lemon slices and 4 tablespoons unsalted butter to the skillet. Stir until the butter melts and the sauce is combined.
Return the mahi-mahi fillets to the skillet and cook for 1 to 2 minutes more, spooning the sauce over the fillets.
Turn off the heat, sprinkle 1 tablespoon chopped fresh parsley over the fillets, and serve immediately.