How does Atticus use his parenting style to instill conscience in his children?
Atticus' parenting style is one that occured possibly out of circumstance, not necessarily choice. He treats his children the best he knows how, but most accurately, he treats them like he treats his job. He is described by Scout as having "courteous detachment" for them, but this phrase has a much deeper meaning that addresses how he parents and why he gets them to develop conscience.
First of all, Atticus is always kind, level-headed, and fair. He tolerates very little of inappropriate behavior as we see in the loss of Jem's pants, the playing of the Boo Radley game, and Jem's destruction of Mrs. Dubose's property. We never see Atticus lift a finger to punish a child, but he uses a firm voice that he likely uses in the courtroom to achieve his means. We do see him expect Jem to pay right for the wrong he did to Mrs. Dubose as Jem is forced to read to her for days and days after he hacked her bushes to pieces.
Atticus doesn't bother himself with great emotion for the children, but does allow each of life's teachable moments to have their center stage at appropriate times. For example, he uses Mrs. Dubose to teach the children courage and that people will do things they don't necessarily mean because they are hurting in another way. He repeatedly uses "walking in someone else's shoes" to teach the children the concept of understanding and tolerating others who are different from us. He makes a deal with Scout to read with him at home tricking her into learning even more than she would have at school. He lives by example. The kids want to know why Atticus won't hunt or go shooting. The kids come to understand through Miss Maudie that Atticus has an unfair advantage over other living things.
Atticus' abilities to lead by example, to use life circumstances to discuss life's mysteries, and to maintain a level-headed, emotionless approach to problems helps the children to develop a strong conscience that understands right from wrong and justice from inequity.