Heat half of the olive oil in a medium–large saucepan over medium heat until shimmering.
Add the anchovy paste, finely chopped garlic, and chopped chili to the pan and sauté for 2 minutes until fragrant.
Add the drained, lightly mashed tuna and sauté, stirring, for about 10 minutes to warm through and combine flavors.
Stir in the canned tomatoes, fine salt, and ground pepper, bring to a simmer, and cook uncovered for 30 minutes, stirring occasionally.
Halfway through the tomato simmer (about 15 minutes in), add the rinsed capers and halved olives to the sauce and continue cooking.
While the sauce finishes, bring a large pot of water to a boil; add 2 tablespoons coarse sea salt, then add the angel hair pasta and cook about 3 minutes until al dente.
Drain the pasta, reserving a little pasta water if desired, then return pasta to the pot.
Toss the cooked pasta with the sauce, add the remaining olive oil if any, and mix until evenly coated, adding a splash of reserved pasta water if needed to loosen.
Sprinkle with finely chopped fresh parsley and serve immediately.