The father of Doodle (referred to as "Daddy") doesn't really play a very major part of the novel. We can imagine his sadness at the birth of Doodle, and we are told that it is Daddy who asks the carpenter to build a coffin, obviously predicting an early death. But also, perhaps to make up for this, it is Daddy who makes Doodle a go-cart. Daddy of course shares a tremendous sense of pride in Doodle's ability to walk.
Interestingly, during the blight of 1918 we receive more information, particularly in his response to what has happened to his crops:
Doodle and I followed Daddy out into the cotton field, where he stood, shoulders sagging, surveying the ruin. When his chin sank down onto his chest, we were frightened, and Doodle slipped his hand into mine. Suddenly Daddy straightened his shoulders, raised a giant knuckly fist, and with a voice that seemed to rumble out of the earth itself began cursing heaven, hell, the weather, and the Republican party.
This shows Daddy in a more human light, especially as we are told that after this Doodle and his brother knew that "everything was going to be alright." It was silence that they feared from Daddy.
It is Daddy who identifies the Scarlet Ibis, and during the meal time conversation Daddy shows preference to Doodle, ignoring the narrator's protestations and denials. It is clear that although he is a minor character in this short story, he plays an important role in providing love and security for Doodle and being a role model for both of the boys.