The problem was that they didn't know what was wrong with Asuna at first and certainly didn't know that she was on ALO, if they did then they probably would have found about Sugou's experiments. even when Yui confirmed she could detect "mommy" up in the tree there was little time before Asuna's and Sugou's wedding. so at that stage Kirito's proof was
a blurry picture which has someone who looks like Asuna, easily explained away by ALO's developer as being a Titania NPC who is apart of the Grand Quest (remember that Oberon's supposed to be waiting at the top and Titania is his wife in Shakespeare's play A Midsummer Night's Dream so it would make sense that Titania would be waiting aswell)
a Navigation Pixie saying that her "mommy" is up in the tree, again could be explained off as a game object's bugged reaction to possible future content given that Kirito's Character is already bugged having it's SAO stats.
given that in the past when the NervGear was removed the player was killed I would think Asuna's parents would err on the side of caution. even if Kirito explained who and what Yui is it, it might be too unreal for them to accept the risk of Asuna's brain getting fried, at least not before the wedding, which I would think Sugou would get called in to give his thoughts, where we know Asuna waking up works against him (remember that she hates him and if she was to wake would call off the wedding).
Now, aside from what I said above, how Sugou could explain away Kirito's evidence that Asuna was trapped in ALO, he also has inside knowledge of the NervGear having not only access to the technology as being the head of Rect Progress but he also worked under Kayaba. He could easily add more worry to Asuna's parents that there could be possible brain damage because of how the NervGear works.
And up until this point there was no indication that the NervGear still wouldn't fry her brain if removed. Clearing SAO only allowed (if not by force) players to log out. While yes, Kirito did die in ALO, this can be argued that Kirito didn't technically die, as his Remain Light was still there and was resurrected by Leafa. One can compare this as being the same as one using the Divine Stone of Returning Soul (resurrect item Kirito got hoping to revive Sachi) while in the Death Game, just with a much longer timer (in ALO it was 10 minutes).