Bring a large pot of water to a rolling boil and add a large pinch of salt.
Add the angel hair pasta and cook according to the package instructions, about 4–5 minutes, until al dente.
Before draining, reserve 3 tablespoons of the pasta cooking water in a small bowl or mug.
Drain the pasta using a fine-mesh sieve or colander.
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 about 2 minutes, stirring, until fragrant and not browned.
Stir in the chopped dried chili (or 1/4 teaspoon chili flakes) and the dried oregano until combined.
Add the drained pasta to the pan with 1 tablespoon of the reserved cooking water and toss to coat in the oil and aromatics.
Add more of the reserved cooking 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 parsley.
Divide the pasta among plates and garnish with grated Parmesan cheese, if desired.