Make the salsa: Blend 1 Tbs of water with the lemon juice, capers, garlic, parsley, oil, and salt in a small food processor or blender until a sauce forms. In a bowl, toss the sauce with the avocado.
Season the tuna with the pepper and grill on a clean, oiled grate over medium-high heat until marked, about 2 minutes. Lift and pivot the tuna 90 degrees and grill for another 2 minutes on the same side. Turn the fish over and cook to desired temperature, another 3-5 minutes for medium. Serve with the salsa.