Heat olive oil in a large saucepan over medium heat. Cook and stir anchovy and capers in hot oil until anchovy crumbles, about 2 minutes. Add garlic; cook and stir until fragrant, about 1 minute.

Pour white wine, oregano, and red pepper flakes into the skillet and increase heat to high; cook until mixture reduces and about 3 tablespoons of liquid remain, 2 to 4 minutes.

Stir tomatoes into the skillet and bring to a simmer. Season tomato mixture with salt, black pepper, and cayenne pepper. Reduce heat to medium and simmer until slightly reduced, about 10 minutes.

Stir tuna and 1/4 cup parsley into tomato sauce, breaking up tuna with a wooden spoon while stirring. Reduce heat to medium-low and simmer for about 10 minutes.

Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until almost cooked through and still slightly firm to the bite, 9 to 11 minutes. Drain and transfer spaghetti back to the pot.

Pour tomato sauce over spaghetti into the pot; stir to combine, cover the pot with a lid, and let sit until spaghetti is cooked through, about 3 minutes. Ladle spaghetti into bowls, drizzle with extra-virgin olive oil, and sprinkle Parmigiano-Reggiano cheese and parsley over the top.