What is the ruling on lying to one's parents after swearing an oath that one's brother is not at home?

Your oath to your brother is a binding oath of righteousness, as long as you did not tell anyone. As for your oath to your mother, it is swearing to a lie, unless it was by way of equivocation or dissimulation. This is permissible if the intention is to divert from the truth without uttering a pure, forbidden lie. If the oath to your mother was a pure lie, then it is a yamin ghamus (a perjurious oath), which is one of the major sins. Sincere repentance is obligatory for it. The majority of scholars hold that there is no expiation for it except repentance, while Imam Shafi'i was of the opinion that expiation is obligatory, and this is more cautious and better.