What does paralleling the adverbial use of Fr. juste mean?
A quick review of the relevant entries in the online Dictionary of Middle English (adj and adv) and the Dictionnaire du Moyen Français shows that the word has pretty much the same wide range of adjectival/adverbial meanings in both languages (but in French it has nominal usages which do not occur in English).
It would take a far more knowledgeable and painstaking scholar than I to sort out the exact history of how these senses evolved and whether influence ever flowed from North to South across the Channel. (It seems unlikely; but this is the era of the Hundred Years' War, when the English occupied large swathes of France, so it's hardly impossible.) I suspect that the phrase "paralleling the adverbial use of Fr." means no more than that nobody competent to do so has yet taken the trouble to find out.
How did the adverb evolve from meaning exactly to merely?
It's not so much an evolution into as an accession of a new meaning. As Peter Shor pointed out in his Comment above, this is a logical extension. OED 1's first citation for its sense 5, "no more than; only, merely; barely", is right on the cusp:
1665 R. Hooke Microgr. vii. 38 Distilled water, that is so cold that it just begins to freeze.
It's a very small step from "exactly to this point, and no further" to "no further than" or "no more than".
1739 Chesterf. Lett. (1774) xxxvi. 125 He can just be said to live, and that is all.
EDIT (at OP's request):
mere(ly) has a very similar history. It derives from Latin merus meaning (of wine) "unmixed" and by the end of the 15th century had come to mean "wholly, utterly", a meaning which endured into the 17th century:
'tis an unweeded garden, / That grows to seed; things rank and gross in nature / Possess it merely —Hamlet, 1601
This sense of mere is still present in fossil phrases like mere folly; but it has mostly ceased to be productive, because it has been superseded by the extended sense of "no more than":
The greatest state we see / At best is merely vanity —Sidney, 1586