Bring a large pot of water to a boil and season it well with salt (it should taste like a flavorful broth).
In a large skillet, heat the olive oil until shimmering. Add the anchovies and step back: they'll splatter! Cautiously stir around with a wooden spoon and then add the garlic, allowing the garlic to color a bit, and finally the red peppers with a pinch of salt.
Cook all together, allowing the peppers to color, but monitoring the color at the bottom of the pan. Lower the heat a bit if necessary. When the bottom of the pan is a dark color, and the peppers have taken on color themselves, add the white wine and use a wooden spoon to scrape up the brown bits. Add the butter and allow the sauce to thicken.
Drop your pasta in the pasta water and cook one minute less than the package directions say. Lift into the pan with the anchovies, garlic, and peppers and add a ladleful of pasta water. Stir all around on high heat until all of the liquid is absorbed.
Turn off the heat and stir the lemon juice or white Balsamic (taste to see if it needs more), then stir in the Parmesan and parsley. Serve with more Parmesan and parsley sprinkled on top.