After a lot of searching, it turns out that the story about Rabbi Akiva's parents is sourced to Rabbi Yosef Mashash's book נחלת אבות חלק ג', pp. 167-171. Rabbi Mashash himself sourced it to the introduction of a book called Ruach Chen (ספר רוח חן), which allegedly brings it in the name of another book:
"והנה סיבת התגיירותו של יוסף אביו של רבי עקיבא, כתב בספר רוח חן בהקדמתו בשם ספר אחד, וזה הוא שנשאר בזכרוני..."
I have only found three books by the name of Ruach Chen - one is a commentary on part of Moreh Nevuchim, attributed to several different authors over the ages, among them Rabbi Yehudah Ibn Tibon, Rabbi Shmuel Ibn Tibon, Rabbi Anatoly (exact identity disputed), and more. The second is a kuntress by Rabbi Noach Gad Weintraub and the third is the name of a commentary on the apocryphal book Wisdom of Solomon, authored by Naftali Hirtz Wessely. The story doesn't appear in any of these, as far as I can tell. I suspect that it might be some manuscript that Rav Mashash had seen at some point, but I can't be certain.
Does anyone have any idea which book it is and whether it can be accessed online?
Side-note: Other than Gershon Bar-Kochva's lecture (see linked question), a couple of other YouTube videos and נחלת אבות, I have yet to find mention of this story anywhere else. And all the videos appear to refer to נחלת אבות explicitly or implicitly. Finding other sources that mention this story might help in finding its original source.
Second side-note: As this is a story related to Rabbi Akiva, I would not be surprised if the Chen part comes from the acronym ח"ן, חכמת נסתר, i.e., that this is some sort of Kabbalistic work. But this is just me speculating.