Drain tuna in a sieve set over a small bowl, squeezing to press out as much oil as possible (reserve the oil). Break up the tuna with a fork and set aside.
Heat 1 tablespoon of the tuna oil in a large nonstick skillet over medium heat. Add garlic and cook, stirring, for 30 seconds. Add tomatoes with their juice and crushed red pepper; break up the tomatoes with a fork. Simmer over low heat until slightly thickened, about 15 minutes. Stir in tuna and set aside.
Meanwhile, cook spaghetti in a large pot of boiling water until just tender, 8 to 10 minutes or according to package directions. Drain it and return to the pot. Add the sauce and 3 tablespoons of parsley; toss to combine. Taste and adjust seasonings. Transfer to a serving bowl, sprinkle with the remaining 1 tablespoon parsley and serve immediately.