People knowing who Jon's parents are does not mean anything for his bastard status. One is born a bastard if he is born out of wedlock, regardless to who the parents were.
Him being son of Lyanna and Rhaegar does not prove that they had married as well because Rhaegar was already married to Elia Martell and traditionally Westeros is a strictly monogamous region by faith and culture. Being a bastard of a Prince does not mean that someone becomes legitimate child. There are plenty of examples of Bastards sired by Targaryen Princes and Monarchs e.g. Daemon Blackfyre, Aegor Bittersteel, Brynden Bloodraven etc.
It is however entirely possible that Rhaegar may have married Lyanna following the precedent of Aegon the Conqeuror who had two wives. King Maegor also had multiple wives. Quoting GRRM:
[Questions concerning Targaryen polygamy.]
Maegor the Cruel has multiple wives, from lines outside his own, so
there was and is precedent. However, the extent to which the Targaryen
kings could defy convention, the Faith, and the opinions of the other
lords decreased markedly after they no longer had dragons. If you have
a dragon, you can have as many wives as you want, and people are less
likely to object.
Simple answer is, from Season Finale, we do not know if they had married or not. Thus we cannot say if Jon would remain a bastard or not.
UPDATE: In Season 7, it is revealed that Rhaegar had his marriage to Elia annulled and married someone else. That someone else can only be Lyanna, making Jon a legitimate Child. So it is all but confirmed in the show that Jon is now a trueborn Targaryen.
Gilly: What does "annulment" mean?
Sam: It's when a man sets aside his lawful wife.
Gilly: Maynard says here that he issued an annulment for a Prince "Ragger"
and remarried him to someone else at the same time in a secret
ceremony in Dorne. Is that a common thing in the south or--?