Is the wife right or wrong in her jealousy over her husband regarding his brother's wife, and what is the proper course of action in this situation?

It appears that your wife erred in writing the message to your brother's wife, and your brother's wife erred by informing her husband, and her husband erred by informing the parents. We advise your wife to apologize to your brother's wife, to strive to reconcile between your wife and your family, to put the matter behind you, and for family members to adhere to the regulations of Sharia in dealing with non-mahram relatives.