Here, Parker (Small Business, 1986) tries to understand a mother he barely knew--in a thin novel whose fictional protagonist remains equally elusive. Noting that his mother disappeared when he was 15, Parker proceeds to tell the story of Austrian-born Anna Moser, who changes her first name each time she changes her life. The daughter of ill- matched parents--a science-fiction writer and a dancer--Anna is a gifted pianist as a child, but by adolescence apparently has lost interest. She becomes a clerk in a fancy Viennese store, along with friend Cybelle, who soon leaves to work as a maid in England. The time is the late 1930's, so Anna, who is Jewish, also leaves, though she never seems to understand quite what is going on. In England--a barely credible and mercifully brief interlude--she becomes Ann, the household maid. Though the Nazis now control Austria, Ann impulsively returns in 1938 to Vienna. There, raped by German soldiers, she flees to old friends, the Hartmanns, whose doctor-son Peter just happens to have American visas for him and Ann. They marry immediately, sail to New York, then soon move on to Florida, where Peter has a job. Peter is very decent but dull, so when handsome refugee David, whom Anna met earlier in New York, turns up, it's not long before she ditches Peter and marries David. Anna works as a demonstration pianist; a son, Will, is born; but by 1950, David has found someone else. Anna, now in her final incarnation, takes up with brutish but generous Jake and becomes Annie, because it ``sounds familiar, fun, perhaps ever exuberant.'' By the close, though, Jake's questionable enterprises, his rough friends and rough ways will all prove too much for Annie. Little from the surface ever connects to substance in a novel that is more incident than insight. Poor ``Anna, Ann, Annie'' is finally An-onymous, unknown and unknowable.