It's not clear to me if they are taking the piss out of the over the top, bought and paid for propaganda documentary that Piers Morgan got ahead of himself with(but how many people actually watched that in the end?) or that they've unwittingly parodied it. Perhaps because that was bought and paid for too? The first half in particular, is not designed to make anyone laugh, it's to confirm their biases and make them nod their heads IMO. As is the second half, focused on the PL being useless and doing nothing but at least they attempted some humour there. You could argue the first half(which is quite long) is a setup for the pay off in the second half but it's not much of a pay off(mildly amusing at best, but maybe I'm biased) and it would have still worked without the first half of the video(somebody really wanted to get all of that in there).
It does suspiciously look like Carragher somehow got an advanced screening of this, maybe all of the people at sky did... and the reaction he had is what they want: "Come on PL! We know they're guilty, you don't have to waste time proving anything, just punish them already".
We already know where Sky's allegiances lie as an organisation. It's not beyond the realms of possibility but it could all be coincidence and something that a Liverpool fan in the Paddy Power offices has been pitching since last year(and forwarded to his mate Carra) I suppose. I still think the overall tone of the video is a little different than their usual content personally.