Bring a large pot of salted water to a boil, then add the angel hair pasta and cook according to package instructions (about 4–5 minutes) until al dente.
While the pasta cooks, heat a large frying pan over medium-high heat and add the olive oil.
Add the sliced garlic to the pan and cook for about 2 minutes, stirring, until fragrant but not browned.
Stir in the chopped dried chili (or 1/4 tsp chili flakes) and the dried oregano, mixing to combine with the garlic oil.
Before draining, reserve 3 tablespoons of the pasta cooking water in a small bowl or mug, then drain the pasta in a fine-mesh sieve or strainer.
Add the drained pasta to the frying pan with 1 tablespoon of the reserved cooking water and toss or stir to coat the pasta in the garlic-chili oil.
Add additional reserved pasta water as needed to create a thick, glossy sauce that coats the pasta.
Season with sea salt and freshly ground black pepper to taste, then stir in the chopped fresh parsley.
Divide the pasta among plates and garnish with grated Parmesan cheese, if desired.