This is indeed what the Shela (R Yeshaya Horowitz, 1565-1630) writes in his work Shnei Luchot Habrit with the exact words you quote in your question (h/t to @ba for identifying the exact source as Mesechet Tamid, Ner Mitzva 100)
Here is the original as reported by Hakolhayehudi (artscroll has a similar version with translation)
לבי אומר שעת רצון לתפלה זו בערב ראש חודש סיון, הוא החודש שבו נתנה
התורה, ואז נקראים בנים לה' אלקינו. וראוי לישב בתענית ביום ההוא הוא
ואשתו, ויתעוררו בתשובה ויתקנו כל ענייני הבית איסור והיתר וטומאה וטהרה
וכל העניינים, ויתנו צדקה לעניים הגונים. ואם אפשר לבעל להתענות אז הפסקה
מה טוב ומה נעים, ועל כל פנים יהיה תענית גמור ככל דיני תענית צבור
The Shlah mentions giving tsedaka (which many institutions who send out the prayer by email don't forget) but I hadn't seen anyone mention that he writes father and mother should also fast !
DailyHalacha expands
There is a special prayer that parents should recite on Ereb Rosh
Hodesh Sivan (the day before Rosh Hodesh Sivan) on behalf of their
children. In general, Ereb Rosh Hodesh is a day of prayer and
repentance, and many people have the custom to fast and recite
Tehillim on this day. Ereb Rosh Hodesh Sivan, however, is especially
designated as a day to pray that one’s children should grow to become
righteous Sadikim and Torah scholars. Since Sivan is the month in
which we accepted the Torah, the day before the onset of this month is
a particularly auspicious time to pray for one’s children’s growth in
Torah.
They bring a scoop however!
Although the prayer is often called “the Shela’s prayer,” it was
actually written by Rav Shabtai Sofer.
but I couldn't find independent confirmation of this.