It was, I suggest, the aftermath that actually provided good grounds. For instance, several 'giants' of literary criticism were demonstrably shown, because they'd ill-advisedly chosen to stray into topic areas which they were ill-equipped to handle, to be talking nonsense. One topic was mathematics where it was shown very clearly that a certain well-known 'intellectual' was manifestly ignorant of the mathematics she enjoyed quoting in support of her 'arguments.
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I don't think there was much demonstrated in the aftermath, just a lot of mudslinging on both sides. While I thought the parody was hilarious, the "serious" followup, in Sokal and Bricmont's book, is not that great. It mostly boils down to: some random critical theorists make too much use of crappy or stretched mathematical/scientific metaphors. That point was demonstrated much more penetratingly in the parody.
The book, though, feels like a literary scholar-pedant finding a metaphorical reference to Faust in a physics paper and launching into a longwinded critique about how the physicist has clearly misunderstood Faust. Well, maybe, and if that were common it'd be an annoying writing style, but errors in interpretation of Faust aren't really the point if it's a physics paper!
And to make matters worse, they don't even go after most of the big names--- there are no chapters in the book on Derrida, Foucault, Althusser, Barthes, etc., because they couldn't find any material to critique.
The problem I have with Derrida, Focault, et. al. is that they're effectively impossible to critique. The density of their writing, combined with their tendency to redefine terms on the fly means that critiques often end up being critiques of interpretation, rather than critiques of the work itself.
Of course, the writers would argue that that's exactly the point. The main idea of postmodernism is that the work is inseparable from the interpretation.