Combine sun-dried tomatoes, 2 tablespoons olive oil, balsamic vinegar, garlic, sun-dried tomato oil, and sugar in a small bowl; set aside.

Remove purple translucent membrane from each fillet using a sharp knife. Cut each fillet, crosswise, into 3 to 4 pieces; season with salt and black pepper.

Heat remaining 1 tablespoon olive oil and butter in a large skillet over medium-high heat. Add monkfish; sear until golden brown, about 3 minutes. Flip; continue cooking until opaque throughout, 2 to 3 minutes more. Reduce heat to low, transfer monkfish to a plate; cover with aluminum foil to keep warm.

Add sun-dried tomato mixture to the skillet; quickly stir until just warmed, about 20 seconds. Spoon over monkfish; garnish with parsley.