Melt 2 tablespoons of the butter in a small saucepan. Add the peas, cover, and simmer over very low heat until tender but still firm, 8 to 10 minutes. Add the cream and simmer until thickened, about 5 minutes. Season with salt and pepper to taste.
Cook the fettuccine in a large pot of boiling salted water until it is al dente. Drain the pasta and transfer to a heated bowl that contains the remaining 1 tablespoon butter. Mix well, sprinkling the Parmesan cheese over the pasta as you toss. Add the peas and cream, mix, and serve at once, with chopped parsley sprinkled over the top.
Note: If using frozen peas, do not thaw them, but add them still frozen to the hot butter. Simmer, covered, until they are tender but firm. This will take a little longer than fresh peas, 12 to 15 minutes in all.