There is little doubt that Atticus's expertise in the law influences his parenting of Scout and Jem.  Certainly, his philosophy of considering things from another's point of view stems from his lawyer's comprehension of the importance of understanding the motivations of others, as does his impartial treatment of others, be they poor or rich.  Scout cannot miss the equity with which he deals with the Cunninghams, Miss Dubose, and others.  His sense of fairness clearly is conveyed to Scout when she overhears her father discusses the upcoming trial of Tom Robinson when Atticus says he cannot face his children if he does not take the case:

I hope and pray I can get Jem and Scout through it without bitterness, and most of all, without catching Maycomb's usual disease. 

In addition to learning from Atticus's exemplary behavior as a just man such as how he never raises his voice to any witness, Scout is also exposed to conversations dealing with legal matters, court cases, and so on.  For instance, little Scout speaks to Mr. Cunningham of "entailment" when she intervenes at the jailhouse where the men hostilely surround her father; having overheard Atticus and Mr. Cunningham's discussions, she thinks this an appropriate topic without fully understanding the meaning of the term. 

Of course, Scout becomes interested in legal matters from this exposure and avidly follows the trial of Tom Robinson.  Several times in her narration of Tom's trial, she alludes to things she has learned from Atticus regarding witnesses:

Never, never, never, on cross-examination ask a witness a question you don't already know the answer to, was a tenet I absorbed with my baby-food.

And, throughout the trial Scout provides not only narration, but analyses based upon her observations.  For instance, as she listens to the interrogations of Mr. Gilmer and of Atticus, she observes,

Slowly but surely I began to see the pattern of Atticus's questions:  from questions that Mr. Gilmer did not deem sufficiently irrelevant or immaterial to object to, Atticus was quietly building up before the jury a picture of the Ewells' homelife.

Clearly, from her father's involvement in the law, Scout's experiences are broader than if she were raised by a man of some other profession, such as an accountant, for example.  For, she witnesses the interplay of society through her vicarious experiences, and she learns to think analytically as her lawyer father leads her to seek truth and understanding in the manner in which he himself does.