Is the brother to marry and move between his home and his sister's home, or is his marriage to be postponed until his sister marries after the death of their parents?

There is no impediment to your brother completing his marriage, and he is not required to stay with you until you get married, because a woman residing alone is permissible if there is no fear of temptation. If it is not safe, then you must reside with one of your mahrams, or a righteous woman should reside with you. Note that it is permissible to offer a woman to a righteous man for marriage.