Heat olive oil in a large skillet over medium heat. Generously sprinkle the grouper fillets with salt and black pepper on both sides and cook until opaque and flaky, approximately 3-4 minutes per side. Remove from heat and cover to keep warm.
Add the butter to a small skillet set over medium heat, whisking continually until the butter melts and darkens in color, approximately 3 minutes.
Remove from heat and whisk in the fresh lime juice. Stir in the fresh parsley leaves and season with salt and black pepper, to taste.