Abner Snopes appears to be a man consumed with anger against others for slights, imagined or real, that causes him to react in an inflammatory way, burning down the property of those who have offended him. For Abner, to be related to him is to support him and to back him up, being complicit in his illegal activity and working against the authorities who are trying to convict him. Thus it is that Abner expects his son, Sarty, to lie for him at the hearing which opens the story. Although he says nothing, Abner recognises that he was going to testify against him. Note what he says to his son that evening:
You're getting to be a man. You got to learn. You got to learn to stick to your own blood or you ain't going to have any blood to stick to you. Do you think either of them, any man there this morning, would? Don't you know all they wanted was a chance to get at me because tey knew I had them beat? Eh?
Thus we see the immense pressure placed on Sarty by his father. To be related to him is, if not to share in his violent tendenies, to be complicit in his violent acts by supporting his father and lying if necessary to protect him. All of which makes his act of defying his father so powerful at the end of the story.