Bring a large pot of salted water to a boil and cook the spaghetti. Meanwhile, in a food processor fitted with the metal blade or a blender, combine the tuna, anchovies, and capers and process.
Add about 1/4 cup of the pasta cooking water and some parsley and process again to make a smooth, creamy sauce.
When the pasta is cooked, reserve some pasta cooking liquid, drain the pasta in a colander, transfer it to the pan with the sauce, and place the pan over medium-high heat.
Toss the pasta briskly with the sauce, and add 1 to 2 tablespoons of the reserved cooking liquid if the pan looks dry.
Transfer to a serving dish. Top with the quartered tomatoes, drizzle with a little oil, and serve immediately.