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Put a pan of water on over a high heat to boil. Meanwhile, snip the chorizo into strips with scissors, and chop the parsley and peppers (check for stray seeds first).
When the water is boiling briskly, add the spaghetti with a good measure of salt, stir and return to the boil. Cook for 3 minutes.
In a large frying pan, heat the oil, add the chorizo and peppers and plenty of black pepper. Cook for a minute or so, until heated through and the juices are stained red from the paprika in the chorizo. Scoop half a mugful of pasta water from the pan, drain the remainder and tip the spaghetti into the frying pan.
Add the parsley and parmesan, toss well and splash in the pasta water, to moisten. Hand round extra parmesan at the table.