Is it foolish to think that Varys, who has spies everywhere and was aware of Rhaegar's intentions to depose his father, knows of Jon Snow's true parentage? There's no textual evidence (that I can think of) to support it (for instance, his conversations with Ned about Robert's bastards), but it seems unlikely that one of the greatest spymaster's in history wouldn't know the greatest secret in Westeros' recent history.
First off I should note I don’t believe Varys was right about “Rhaegar’s intentions to depose his father” – I think he was deliberately stirring shit in House Targaryen as part of his Blackfyre restoration plans, and the supposed secret backer of the Harrenhal tourney being Rhaegar is a red herring placed in TWOIAF on purpose. (I think there’s a lot of things Yandel is very wrong about in TWOIAF, and while some like magic and the Others and such are obvious, there’s many political mistakes and oversights too. He’s a very unreliable narrator, deliberately so.)
Secondly, Varys is an excellent spymaster, but he doesn’t know everything. He only knows what his little birds can see or hear. (Like, Olenna had loud singing cover up her conversation with Sansa, Littlefinger had Sansa meet Dontos in the godswood specifically to avoid the little birds, and that’s why LF reported the Tyrell betrothal plan to Tywin while Varys hadn’t heard anything about it. Also Varys doesn’t seem to have had any awareness of Sandor and Sansa’s interactions.) Note that Varys was still fairly new to Westeros at the time of Robert’s Rebellion and almost certainly didn’t have a full spy network yet, certainly not in the Red Mountains where the Tower of Joy was, and probably not in Starfall either. Jaime relates that during the Rebellion “Prince Rhaegar could not be found”, which means people were looking but couldn’t find him, which means Varys didn’t know either. (Or if he did know, he wasn’t saying, again to stir shit.) The Tower of Joy was very out of the way.
But still, it’s possible that Varys does know about R+L=J, either from his spies back when or gathered knowledge since then. If so, he’s likely factored in the possible revelation of Jon’s parentage into his plans to bring Aegon to the throne. Whether Varys has false counter-information ready in case of a reveal, or if he’d simply be using a semi-truthful legal strategy (Aegon is older; even if Rhaegar married Lyanna it was bigamous and thus illegal; etc), or even if he doesn’t know and finds out when the secret is made public (if it’s made public), it probably doesn’t matter. Varys is very utilitarian: he’ll do what he must, whatever it may be.
(btw I don’t mind if someone asks the same question of all the meta people on tumblr, presumably in hopes of getting different answers, but personally I prefer if it’s proceeded with a statement that that’s what you’re doing. It makes me feel less like a meta monkey dancing for your entertainment, thanks.)