Parshat Haman is a short parsha Esther 3:1 - 7. The parsha introduces Haman and the key part of this parsha refers to how the righteous Mordechai disobeyed the king's order to bow down to Haman.
The key verse is Esther 3:4:
וַיְהִ֗י באמרם [כְּאָמְרָ֤ם] אֵלָיו֙ י֣וֹם וָי֔וֹם וְלֹ֥א שָׁמַ֖ע
אֲלֵיהֶ֑ם וַיַּגִּ֣ידוּ לְהָמָ֗ן לִרְאוֹת֙ הֲיַֽעַמְדוּ֙ דִּבְרֵ֣י
מָרְדֳּכַ֔י כִּֽי־הִגִּ֥יד לָהֶ֖ם אֲשֶׁר־ה֥וּא יְהוּדִֽי׃
Now it came to pass, when they spoke daily unto him, and he hearkened
not unto them, that they told Haman, to see whether Mordecai’s words
would stand; for he had told them that he was a Jew.
From here, we learn that it was because the king's servants spoke to Mordechai daily and each day he refused to listen to them, that it was in this merit that Mordechai and the Jews were eventually spared. (Had it ben just a single or occasional warning, it would not have demonstrated Mordechai's merit as strongly.)
We see at the end of the Megillah, that Mordechai became wealthy as he was appointed vice minister to the king, and he wore royal garments, etc.
So this is why it is a good segulah to read this parshat Haman which refers to Mordechai's refusal to Haman.