Whisk first 8 ingredients together in a medium bowl to combine. Gradually whisk in oil until mixture thickens. Stir in celery, shallots, green onions, pickle, and parsley. Season to taste with salt and pepper. Makes about 2 ½ cups.
Place crabmeat in a large bowl. Mix in enough dressing to coat, about half. Puree remaining dressing in a blender or food processor until creamy. Cover crabmeat and pureed dressing separately and refrigerate at least 1 hour. (Can be made 1 day ahead.) Just before serving, gently stir in cubed avocado. Either stir in some of the remaining dressing or pass separately. Serve Crab and Avocado Remoulade in hollowed out buns, peppers, avocados, or simply on a plate.