In a large pot of boiling salted water, cook pasta 5 minutes less than package instructions for al dente; add cauliflower. Cook until pasta is al dente and cauliflower is tender, 5 minutes. Reserve 1 cup cooking liquid. Drain pasta mixture; set aside.
In same pot, heat oil over medium; add garlic, red-pepper flakes, and anchovies. Cook, breaking up anchovies with a spoon, until garlic is fragrant, about 1 minute.
Add pasta mixture, cheese, and 1/2 cup reserved cooking liquid; season with salt and pepper. Toss to coat, adding more cooking liquid if necessary to thin sauce. Add parsley; toss. Serve, topped with additional cheese, if desired.