Why did Holden's parents want to have him psychoanalyzed?
Holden's parents have to have him psychoanalyzed; he has a nervous breakdown following his chaotic weekend in New York. His behavior leading up to the break-down is critical to their decision.
Holden himself admits on page one of the book that he had a mental breakdown.
"I'll just tell you about this madman stuff that happened to me around last Christmas just before I got pretty run-down and had to come out here and take it easy." (Salinger)
He left school without permission and traveled to New York and did not let his family know that he was home. He checks into a cheap hotel, gets beat up by a pimp, sneaks into his own apartment to see his sister and sneaks out before his parents see him. He hides from them.
He seeks comfort at the home of a former teacher and thinks that he is making sexual advances at him, so he runs out in the middle of the night and ends up sleeping in the train station.
Holden fantasizes about his death, sitting in the freezing Central Park almost wishing that he freeze to death.
He tells his sister that he intends to hop on a bus and go west to work on a ranch.
Holden's parents do not understand their son. Perhaps they are still dealing with the death of Allie, but they are aloof and cold. Their method of dealing with Holden is to keep sending him to different boarding schools instead of keeping him home and talking frequently with him about his feelings. They mistake the pains associated with growing up with psychosis and sending Holden to a psychiatrist is another way of pushing his problems onto someone else. Granted, his behavior is bizarre, but it didn't start that way. But, no one has taken the time, except Phoebe, to get to know and accept Holden. Like the ducks, he is frozen out of his parents' lives and their solution is to bring in a "professional" to "fix" him.
Holden's parents want him psychoanalyzed because after his brother Allie's death he couldn't cope with the event. His brother Allie's death caused him so much despair he coped with it by breaking all the windows in the garage with his bare hands. This was definitely a warning that Holden needed help in order to deal with his brother's death.