Bring a large pot of salted water to a boil: add 2 tablespoons coarse sea salt when boiling and keep boiling until pasta is ready to cook.
Heat half of the olive oil (about 2 tablespoons) in a medium-to-large saucepan over medium heat.
Add the anchovy paste, finely chopped garlic, and chopped chili to the warmed oil and sauté for 1–2 minutes until fragrant but not browned.
Stir in the drained, lightly mashed tuna and sauté for about 10 minutes, breaking up any large pieces.
Add the canned tomatoes, bring to a gentle simmer, and cook uncovered for about 30 minutes, stirring occasionally.
About halfway through the tomato cooking time, add the rinsed capers and halved olives to the sauce and continue simmering.
When the sauce is nearing completion, cook the angel hair pasta in the boiling water for about 3 minutes or according to package directions until al dente, then drain.
Toss the drained pasta with the sauce, the remaining olive oil, and adjust seasoning with the 1/4 teaspoon fine salt and 1/8 teaspoon ground pepper as needed.
Sprinkle the chopped fresh parsley over the pasta and serve immediately.