In medium bowl, combine all dressing ingredients except oil; beat with wire whisk until blended. Add oil; beat until well blended. Set aside.
To section grapefruit and oranges, cut or peel each fruit down to the flesh, leaving no white pith. Between walls of membrane, cut segments straight down with sharp knife. Gently remove segments. Peel, halve and pit avocado; cut into slices about same size as grapefruit and orange sections.
To assemble salad, place shredded lettuce on large platter. Arrange grapefruit and orange segments over top. Toss avocado slices in dressing to prevent darkening. Place avocado on top of fruit. Spoon dressing over salad. Sprinkle with parsley.