Melt the butter in a saucepan over medium heat. Whisk the flour into the melted butter until the mixture is smooth. Slowly beat the half and half into the sauce until completely incorporated. Stir the garlic, parsley, and Parmesan cheese into the sauce, whisking continually. Bring the sauce to a simmer; cook, stirring regularly, until the sauce has thickened, 4 to 5 minutes. Use immediately or refrigerate. You may need to thin the sauce out with more half and half or a bit of milk if you refrigerate it. Serve over hot alfredo noodles.