The theme of Papa's Parrot involves the effects of coming of age in young people, and the way that we tend to want to separate from our parents. It also points at the fact that, once we do separate from our parents both mentally and physically, it takes maturity to realize that this is a behavior that does not come from the heart, but from an inner desire to grow and become independent. Either way, it is a sad realization that sometimes comes to us a little too late in life.
Just take a look at the symbols in the story. The fact that Harry's dad was "fat" and worked in a candy store denotes the innocence of Harry, who had no issue with it, and whose friends also visited the candy store...when they were very young.
Later on, as things become more sophisticated and aesthetic in the eyes of the growing youths, the candy store becomes instantly associated with childhood; this is why Harry and his friends quit going to the candy store and prefer to use the newer, "cooler" penny machines to get their candy from it; they are still young, but now they are growing. Everything associated with their childhood is to be rejected, and the people associated with it, as well.
When Harry's dad gets the parrot, it is clear that he does it because he felt lonely. In fact, he speaks to it more than he does to people. When he has his heart attack, Papa can no longer tend to the candy shop, so Harry steps up and promises to take care of it. It is then when he hears the parrot speak the words that he hears his owner say the most:
Where is Harry? I miss him.
Feeling pained and guilty, Harry realizes that his Dad loves him and misses him. Harry breaks down and cries, saying, maybe a little too late,
I am here.
Therefore, the key theme is family, its dynamics, and coming of age. It is very hard for parents to see their children grow and leave the "nest". While their children do not realize it, that need to separate hurts parents sometimes at the very core. That is one of the "growing pains" of life.
The answer to this question can be found in the response of Harry when he goes to his father's shop and hears the words that Rocky, the parrot that belongs to his father, utters to him. The fact that these words must have been said many times for the parrot to learn them makes their import even more significant. Harry hears Rocky say "Where's Harry?" and then "Miss him! Miss him!" This causes Harry to experience intense guilt and sadness as he realises for the first time just how much his father loves him and has missed him whilst he had been desperate to avoid him because he was embarrassed by him.
The story's theme therefore is two-fold: firstly it focuses on the way that time is so short and we must make the most of the relationships that we have now rather than ignoring them. Secondly, it also comments upon the tendency of adolescents to move too far away from their parents and to miss out on that crucial relationship because they are going through a stage where they choose not to associate with their parents at all.