Put the parsley, green onions, garlic, anchovies, salt and pepper into a food processor and chop into a puree, stopping to scrape down the side of the processor bowl and top. Add two tablespoons of water and process until the mixture is nearly smooth.
Scrape the puree into a non-metallic bowl and add the remaining ingredients. Blend with a whisk until uniform in texture. Whisk in a little bit of water if needed to bring the sauce to a very thick but pourable texture. Refrigerate.