Scrub mussels under cold running water and pull off any dark beards. Discard any with cracked shells or ones that stay open when tapped.
Bring a 6 quart / 5.7 L pot of generously salted water to a boil for the spaghetti.
Warm olive oil in a 30 cm / 12-inch skillet with a lid over medium heat. Add sliced garlic and chili flakes, cook 1 minute until fragrant but not browned.
Pour in the white wine and bring to a simmer, then add the mussels. Cover and steam 5 to 6 minutes, shaking the pan once, until the shells open.
Lift the mussels into a bowl with tongs, leaving the liquid in the skillet. Discard any mussel that hasn't opened after the full 6 minutes. Once cool enough to handle, shell about two-thirds, leaving the rest in the shell.
Drop the spaghetti into the boiling water and cook 2 minutes short of the package's al dente time. Scoop out 1 cup / 240 ml pasta water, then drain.
Add the drained spaghetti to the mussel liquid in the skillet with a splash of reserved pasta water. Toss over medium heat for 2 minutes, adding more pasta water as needed, until the pasta is coated and al dente.
Return the shelled and unshelled mussels to the skillet with the chopped parsley. Toss for 1 minute to warm through.
Season with salt and black pepper. Plate straight from the skillet and top with more parsley.