Howland Reed is alive. He is the last survivor of the Tower of Joy and knows what transpired there. He has kept this a secret due to his loyalty and friendship with Eddard Stark.
But as you correctly note, cynical lords won't believe word of one man. Outside North it would be even more complicated because Reed is a Crannogman who are not viewed favorably in the South.
Besides, If Jon is a Targaryen and son of Prince of Dragonstone, Rhaegar Targaryen, that's a concern for whole realm, not just the North.
By law, (If Lyanna and Rhaegar married,
which we do not know they did, see update below.), a grandson (Jon) by eldest son (Rhaegar) comes before a daughter (Daenerys) in succession for a King (Aerys II). If Jon is not a bastard, he is infact the rightful King of Westeros as heir of the mad King, not just North.
Even though there is no Polygamy in faith of the seven, Aegon the Conqueror and Maegor the Cruel practiced it. 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.
So Rhaegar could be married to both Elia and Lyanna simultaneously.
There is no tangible proof for Jon's parentage because only these people were present at the tower or knew about it or possibly knew about it:
- Eddard Stark (Dead)
- Howland Reed (Alive)
- Lyanna Stark (Dead)
- Arthur Dayne (Dead)
- Gerold Hightower (Dead)
- 4 unnamed Northmen (Dead)
- 2 unnamed Dornish girls (Fate unknown)
- Oswell Whent (Dead)
- Rhaegar Targaryen (Died at Trident before Tower of Joy Showdown)
- Elia Martell (Died at KL before ToJ showdown, She may not have known anything)
(One can argue that two other members of KG namely Prince Lewyn Martell and Ser Jonothor Darry may have known as well but it appears Prince Rhaegar did not confide in all the KG. Ser Barristan Selmy and Ser Jaime Lannister did not know about it despite being members of KG. In any case, Both Lewyn and Jonothor died as well)
Another thing, unrelated to the show but relevant to the books, is George has stated that Meera and Jojen Reed might know something about what transpired in the Tower of Joy.
Anyways, Since all other witnesses are dead, Even if the two girls are alive and add their voice to Reed's, that still won't be enough proof. Besides, Jon does not look like your typical Targaryen but that is the case with many of the half-Targaryens over the history e.g. Prince Baelor Breakspear, Prince Valar, Prince Daeron the drunken, Aegor Rivers etc.
Eddard had fabricated the story of fathering Jon on a common woman named Wylla. If she can be found, investigating her would also prove Jon is not son of Eddard Stark but her whereabouts and status is unknown in Show, while in Books she was a serving woman in Castle of Lord Dayne.
To prove his Targaryen blood, Jon will have to do something which only a Targaryen can do. That is, Riding a dragon. As it happens, Daenerys is on her way to Westeros with her Dragons. Drogon is claimed by Daenerys but Viserion and Rhaegel are up for the taking. If Jon manages to claim one of them, no one will be able to doubt his Dragon blood.
There is also the the possibility of Bran adding his voice to Reed's. The whole issue will come up only if Bran does that. Otherwise, Jon is now the White wolf for the realm, not the white Dragon. Bran is not just a boy, If he returns to inform the realm about this, he is the eldest surviving son of Lord Eddard Stark,
thus rightful King in the North. Lords have given their allegiance to a bastard instead of a trueborn Stark female, but will they keep Jon as King when their trueborn male Stark liege lord returns? Would Jon want to keep the seat which is rightfully Bran's? I don't believe so. Apparently, as of Season 7, Bran doesn't want to take his rightful place and Jon remains King in the North.
TL;DR No there is no other proof of Jon's parentage except Reed's word.
UPDATE: As of Season 7 Episode 5, it has been confirmed that Rhaegar's marriage to Elia was anulled and he married another which can be no one other than Lyanna. That makes Jon a legitimate Targaryen and therefore the rightful owner of the Iron Throne of Aegon the Conqueror. The records of the High Septon can be produced as evidence for Lyanna's marriage to Rhaegar but that still won't prove Jon is their child.