Bring a large pot of salted water to a boil. Meanwhile, heat oil in a large saucepan over medium heat. Cook shallot, stirring occasionally, until softened, 3 to 5 minutes. Stir in chile flakes and 3/4 tsp. salt; whisk in prosecco and crème fraîche. Bring to a boil over medium-high heat and reduce to 1 3/4 cups, 10 to 15 minutes.
Gently stir lemon zest, 2 tbsp. juice, and the crab into prosecco sauce and remove from heat. Boil pasta until just tender, 2 to 5 minutes.
Drain pasta, return to pot, and gently toss with crab sauce and parsley. Taste; add more salt and lemon juice if you like.