This is definitely not a poem, but I do believe a great parallel text for The Kite Runner would be A Separate Peace. Compare Gene to Amir and the theme of envy.
You also might look at "Change" by Letitia Elizabeth Landon; it deals with two boys who are very close in their youth, but drift away as they get older and their priorities change.
"A Poison Tree" by William Blake is a great poem to relate to The Kite Runner. There is a link about the author below, followed by a link to the poem. It tells what happens when anger ruins a friendship. Anger is a kind of poison that can start small, like a seed, but if you feed it, it will grow into a tree. I think that's what happens to Amir in the novel, an anger and envy with Hassan that he feeds and feeds until it grows quite large. In the poem, the friend is killed by the anger, and of course, the actions of Amir, you could argue, based on his anger and envy, set the stage for Hassan to die. The poem and the book are powerful lessons about anger.