Bring a large pot of salted water to a boil. Add the angel hair pasta and cook until tender but still firm to the bite, stirring occasionally, about 6 to 8 minutes. Drain, reserving 1 cup of the pasta water.
Stir the oil, parsley, lemon juice and lemon peel in a large serving bowl. Add the cooked pasta and toss with enough reserved pasta water, 1/4 cup at a time, to moisten. Season the pasta with salt and red pepper flakes, to taste. Sprinkle grated lemon peel over pasta for extra flavor and texture. Sprinkle with the Parmesan and serve.
Combine the oil and crushed red pepper flakes in a heavy small saucepan. Cook over low heat until a thermometer inserted into the oil registers 180 degrees F, about 5 minutes.
Remove from heat. Cool to room temperature, about 2 hours. Transfer the oil and pepper flakes to a 4-ounce bottle. Seal the lid. Refrigerate up to 1 month.