The ibis is a parallel for Doodle. Doodle is different from everyone around him. His parents accept and love him but he is certainly different from his brother and other children his age. Doodle doesn't seem to mind this, asking if it matters if he will be different from the other kids at school. Brother doesn't want Doodle to be different (for his own personal reasons more than for Doodle's own sake of fitting in). Doodle is different and therefore is like the ibis: out of place. The ibis is an exotic bird from South America. The boys' father supposes that it was carried by a storm. Being in a strange environment, the ibis cannot survive and falls to its death. Like the ibis, Doodle has had to struggle to survive in his own environment and eventually he also dies because he cannot keep up with Brother's training regimen designed to make Doodle fit in with the other kids at school. Doodle and the ibis find themselves in an environment where it is difficult and/or impossible to survive/fit in.
When Brother finds him dead, the red blood flowing from Doodle's mouth symbolizes the scarlet feathers of the ibis. Brother directly makes the connection between Doodle and the ibis at the end of the story:
For a long time, it seemed forever, I lay there crying, sheltering my fallen scarlet ibis from the heresy of the rain.