Well, let's list what would actually be functionally different. If they admit that their sick, then the gang goes off with their blessing, maybe the adults could provide foreshadowing to the trials ahead, and they'll at least have directions from the valley to the Land of the mists. It's simple and direct, gets the journey started quickly, and is probably the more in-character option of the two.
Now, consider the other option: what If they steadfastly refuse to admit that they're sick, no matter how blatantly obvious it is? Well, that begs the obvious question: why? A possible reason could be that the adults know the gang will go searching for a cure, and-- if we do decide this is the same disease as the one from movie four-- they'd know the gang would try to go get some night flowers from the land of mists. Now, considering the gang has sucessfully done this before, the only reason I can think of that they'd try to hand-wave their illness is if they knew something terrible was happening over at the land of mists at the time that they'd rather risk the illness than let their kids go over there. If that were the case, the gang would leave without their blessing, which can open the possibility of the adults finding indirect means of stopping them: maybe enlist help to go and fetch them before they reach the land of mists. That would have someone well meaning but antagonistic hounding them on their journey to get the night flower, and would emphasize the desperate lengths the adults want to go to keep their kids out of the land of mists.
... Or something like that
whatever the route afterwards, we can assume that the adults not admitting their illness would create some sort of complication for the gang. So the the question is: straightforward and simple, or more complicated and dynamic?
I for one, don't particularly care. I'll follow the route with the best story, and either route can be worked into something interesting.
