Is the questioner considered sinful for severing family ties and accountable for his parents' sadness because his brother did not accept his apology after the estrangement?

You did well by apologizing to your brother, and the sin falls upon him if he continues the boycott. It is not permissible for a believer to abandon his brother for more than three nights, and the better of the two is the one who initiates the greeting. There is no sin upon you for your parents being hurt. We recommend that your brother be advised by someone whose words are likely to influence him, and that the parents' sorrow be a motivation for him to end the boycott, while also emphasizing the importance of accepting the excuse of one who apologizes.