Remember that the theory about Jon's parentage, is still a theory.
From the books, Aemon Targaryen was not aware of Jon Snow's parentage.
There is a comprehensive list of who knows about Jon's parentage in Who knows the truth about Jon Snow's parentage (father & mother), in which Aemon does not appear as a likely candidate.
In reference to Azor Ahai, Aemon has a realisation that it must be Daenerys, meaning that he believes her to be the only possible option:
On Braavos, it had seemed possible that Aemon might recover. Xhondo’s talk of dragons had almost seemed to restore the old man to himself. That night he ate every bite Sam put before him. “No one ever looked for a girl,” he said. “It was a prince that was promised, not a princess. Rhaegar, I thought... the smoke was from the fire that devoured Summerhall on the day of his
birth, the salt from the tears shed for those who died. He shared my belief when he was young, but later he became persuaded that it was his own son who fulfilled the prophecy, for a comet had been seen above King’s Landing on the night Aegon was conceived, and Rhaegar was certain the bleeding star had to be a comet. What fools we were, who thought ourselves so wise! The error crept in from the translation. Dragons are neither male nor female, Barth saw the truth of that, but now one and now the other, as changeable as flame. The language misled us all for a thousand years. Daenerys is the one, born amidst salt and smoke. The dragons prove it.” Just talking of her seemed to make him stronger. “I must go to her. I must. Would that I was even ten years younger.”
-A Song of Ice and Fire: Book Four - A Feast for Crows, Chapter 35 (Samwell IV).
There are also many situations where Aemon speaks with Jon and tells him things in reference to his family, The Starks - and not anyone else - not even a single allusion to anything extra.
In fact, the most recent discussion they have is one in which Aemon give Jon some advice about how to 'rule' succesfully as Lord Commander of the Night's Watch:
Aemon’s final words. “Allow me to give my lord one last piece of counsel,” the old man had said, “the same counsel that I once gave my brother when we parted for the last time. He was three-and-thirty when the Great Council chose him to mount the Iron Throne. A man grown with sons of his own, yet in some ways still a boy. Egg had an innocence to him, a sweetness we all loved. Kill the boy within you, I told him the day I took ship for the Wall. It takes a man to rule. An Aegon, not an Egg. Kill the boy and let the man be born.” The old man felt Jon’s face. “You are half the age that Egg was, and your own burden is a crueler one, I fear. You will have little joy of your command, but I think you have the strength in you to do the things that must be done. Kill the boy, Jon Snow. Winter is almost upon us. Kill the boy and let the man be born.”
-A Song of Ice and Fire: Book Five - A Dance With Dragons, Chapter 7 (Jon II).
Note: this is the same "Egg" or Aegon V Targaryen
was talking about in his fever dreams in Season Five, Episode 7 (The Gift
) of Game of Thrones
From what I know and have seen, there has been no communication between the possible 'knowers' of the truth and Aemon. In fact, it would be a complete waste of time, since anyone who knows Aemon would know that he is wholly devoted to The Watch, and would want nothing to do with the ongoings of The Realm.
Why would anyone tell him anything of the sort? "Hey Aemon, not sure if you're interested but you know that bastard kid from Winterfell? Well, he's actually ... And if he ever decides to join The Watch, just look out for him ok?" <-- doesn't make sense.
The only nugget in all of this is in fact Benjen Stark. He may or may not be aware of more than he has let on. In fact, we don't know exactly why he joined The Watch:
- Could it have been because of the secret(s) in his family? Maybe.
- Could and would he have told anyone, including Aemon? Maybe, but I'd doubt it! If he's taken The Black to escape it all, why would he take the secret(s) with him?
Though Aemon grew fond of Jon and defended as well as helped him, he was not the only person to take a liking to Jon:
At this point, there is no evidence to suggest that Aemon knows about Jon's true parentage.