As far as we know, Tyrion doesn't know this information, and we have every indication that Ned Stark kept this secret to himself only. But! One of the biggest secrets in Westeros before this was Joffrey's (and Tommen's and Myrcella's) parentage. No one told Ned that Jaime and Cersei had incestuously created the "heirs" to the throne; he worked it out for himself by studying texts (and using his brain to figure out the Jaime/Cersei connection).
Another character who is well-versed in the history of the Seven Kingdoms, and always looking to learn more, is Tyrion. Now that he's the Hand of the Queen for Daenerys, he'll surely be even more curious about the Targaryens. And now that Dany, Jon, and Tyrion have met in person, there is even more reason for Tyrion to explore more about Jon. Again, we know there won't be anything in any Westerosi book declaring Lyanna and Rhaegar as Jon's parents, but if anyone can look into it and put the pieces together, it's Tyrion. If scores of Game of Thrones fans could figure it out before it was revealed, surely one of the smartest men in the Seven Kingdoms could too.
