Add the egg yolk, mustard, and lemon juice to a blender and blend until combined to make the aioli. With the motor running at medium speed, very slowly drizzle in the olive oil. The olive oil should be poured in at a dribble at first, just a few drops at a time, and continue in a very slow, thin stream.
Blend until the mixture is creamy and emulsified. Pour the mayonnaise out of the blender and fold in the prepared horseradish and chopped chives. Add parsley for garnishing.
Taste and season with salt and pepper. Refrigerate until you're ready to eat. (The mixture may get thicker as it chills. Consume any leftover aioli within a week.)