The parents are being supported in completely the wrong way and if CA had even an ounce of sense to share between them, they would realise this.
Call it mob mentality, sheeple, opening the lid to the idiot box, etc etc, but what you've essentially got is a family who have whipped up a shit storm and presented some carefully chosen information (and some outright lies) regarding their very ill little boy to a bunch of morons. CA are obsessed, but ultimately, despite what they think, don't know, care about, or love Charlie. A bored bunch of idiots who aren't capable of realising what damage they are doing to the family and the hospital and ultimately a defenceless child.
As to the motivation of the parents, they probably do genuinely believe they are doing the right thing for Charlie but I suspect they have lost all perspective and have convinced themselves they know better than the doctors. Quite clearly they don't, but they are being supported in the wrong way by the wrong people and are unlikely to be capable of being persuaded otherwise. I don't think they are wilfully lying about GOSH, I think they are deluded and there probably isn't anyone in their 'inner circle' who is able to make a breakthrough and convince them of the truth.
Sadly I suspect that no matter what happens with this latest court hearing the family will always come up with a reason why GOSH are lying and why the life support should be maintained. I don't know how educated the parents are (or even if that would make a difference) but I just don't see anyone being able to get through to them.