In a blender, combine 3 tablespoons of the olive oil with the parsley and process until smooth. Season lightly with salt. Transfer 2 tablespoons of the parsley oil to a small bowl. Add the flaked tuna, the mayonnaise, water, capers, lemon juice, anchovies, garlic and the remaining 1 tablespoon of olive oil to the blender and puree until creamy. Season the tonnato sauce with salt and pepper.