I thought only to see the future was heresy, not visions generally. So may be the people of Vorin nations think about visions = visions of the future because of their past (Hierocracy). But visions do not necessary show the future as seen in Dalinar's visions: he's shown the past to take lessons from his visions. And so his visions are no heresy, I think.
But if the Parshendi's hearts (you mean the "real hearts" here, not the gemhearts?) would be infused, wouldn't this be vital to them or at least wouldn't loosing this Stormlight make them weak or less combative?
I didn't see them dieing, weakening or getting less combative in the fight, when Kaladin and Bridge Four stayed behind and joined battle after Sadeas left Dalinar alone.
And if the Parshendi "have" gemhearts as part of their body and are infused with Stormlight, why do the male fighters need to put gemhearts in their beards?
The Parshendi "stopped fighting" and said something but they didn't abandon fight.
In general -- call me naive -- I don't see honorable behavior in the Parshendi. In the first place I don't see something honorable in assassination. And I'm pretty sure the Parshendi are the orderer of Szeth for killing Gavilar. While reading through this thread I wavered about this, but then I'd asked myself why the Parshendi should fight this war if they didn't cause for it. Why not try to convince Elhokar that they did not order the assassination of his father? Only because of the gemhearts? I can't believe that.
And at last: I think that the Parshendi are part of the Voidbringers but (actually) not the Parshmen. If Jasnah expects being the Parshmen Voidbringers, why return to the Shattered Plains (when I remember right, this is her intention after revealing her theory) and not share her wisdom with Dalinar via Spanreed (or even give him a hint) while trying to tell or at least warn Taravangian that there is a potential endangerment all around? (okay, have to admit that there may be reasons not to do this)