Slice half the lemon thinly. Juice the remaining half to yield about 1 1/2 tablespoons juice; set both aside.
Pat mahi mahi dry. Season all over with 1 teaspoon salt and a few grinds of pepper.
Heat oil in a large cast-iron or nonstick skillet over medium-high until just starting to smoke. Sear mahi mahi undisturbed about 4 minutes, then flip and cook 2-4 minutes more until just cooked through. Transfer to a serving platter.
Reduce heat to medium-low. Add lemon juice, garlic and remaining 1/4 teaspoon salt, scraping up browned bits.
Add lemon slices, then stir in butter one piece at a time, waiting until each is almost melted before adding the next.
Remove from heat, stir in parsley, taste and adjust salt and pepper. Pour sauce over mahi mahi and serve immediately.
Notes
Leftovers can be refrigerated in an airtight container up to 2 days.