What insight do we get into Atticus's parenting skills in this chapter of To Kill a Mockingbird?
You don't specifically mention the chapter to which your question refers, but Atticus displays a concerned attitude toward his children and wise nurturing skills throughout the novel. In Chapter 3, Atticus faces a dilemma when he is accused of poor teaching skills by Miss Caroline and a daughter who no longer wants to attend school. Atticus solves the problem by getting Scout to commit to a compromise: She will return to school, and Atticus will continue to read to her each night--without Miss Caroline's knowledge. Atticus also explains to Scout about the importance of not pre-judging people too quickly, that
"You never really understand a person until you consider things from his point of view--until you climb into his skin and walk around in it."
In Chapter 4, Atticus discovers his children playing their new Radley Game; he disapproves of it, but Atticus accepts Jem's word that it has nothing to do with the Radleys (knowing full well Jem is lying) to see if his warning is enough to put a stop to it. In the following chapter, Atticus catches the children red-handed trying to deliver Boo a note--on a fishing pole. Atticus uses his lawyering skills to get Jem to unknowingly admit to playing "an asinine game he had seen us playing," and making Jem realize "he had been done in by the oldest lawyer's trick on record."
Atticus always leads by example, allowing Scout to overhear his discussion with Uncle Jack at Christmas in order for her to understand his reasons for deciding to take the risky step of defending Tom Robinson.
"... do you think I could face my children otherwise?... I just hope that Jem and Scout come to me for their answers instead of listening to the town. I hope they trust me enough..."
He teaches them about humility by not revealing his secret skill--that of being the greatest marksman in the county--and Jem understands that this is one skill of which Atticus takes no pride. It makes Jem even prouder of his father:
"Atticus is a gentleman, just like me!"
Though Atticus is a busy man, attending to his legal profession and representing Maycomb in the Alabama legislature, Atticus spends as much time as possible with his children. He reads to Scout every night, comes home to eat lunch with them, and is always ready to answer even their most difficult questions. He "is never too tired to play keep-away," but he draws the line at playing tackle football with Jem. Atticus is wise enough to realize that his children still need a woman's touch, and that is where Calpurnia comes him. He knows that "I couldn't have gotten along without her all these years," and that Cal will "never let them get away with anything... the children love her."