Orach Chaim 39:2 - Biur Halacha seems to say that a minor can not do any part of making of the Tefilin. However a reading of the Mishna Berura 10 seems to say that a minor can work on producing the parchment. Recently I saw a sign posted in a store in Yerushalayim that was saying that the recent trend to allow boys who are becoming Bar Mitzva to place their own Parshios into the Batim is against Halacha as all parts of the Tefilin making must be done by a adult. Well Mishna Berura allows a child to produce parchment and does not mention this prohibition of a minor installing the Parshios. It mentions not allowing a minor to make the Batim, the Shin, the sewing, and the knots. What is the source that this is forbidden or is there a source that allows it?
The source for this appears to be Gittin 45b (translation from artscroll)
A Torah scroll, tefilin or mezuzot written by [...] a minor [...] are invalid for it is said "and you shall bind them ... and you shall write them" (Deuteronomy 6:8-9) -- whoever is involved in the mitzvah of binding tefilin on his arm is eligible to engage in the writing of [...] tefilin and whoever is not involved in the mitzvah of binding is not eligible in the writing.
So it appears to be a question of definition of what is writing tefilin and how much of the production is called writing.
For what is worth when I bought tefilin for my son from Tefilin Beth El (a well-known tefilin factory near Jerusalem with plenty of haskamot), they allowed my son to start putting the parchment in the batim but asked me to complete it to avoid the issue you mention.
Also seems like the Mishna Brura you cite says a minor can produce the parchment but only under supervision by an adult and lshem mitzvat tefilin.