The Pope demanded from King John to reject the Magna Charta and all parties did not take that document as serious as the second half of the 8 centuries, when it was indeed immensely important&influential (cf. Macaulay, his grandnephew Trevelyan, Maitland a.s.o.).
I think John had hoped that it would just sort of go away once he agreed to it, which it didn't, but his consent to it was more or less forced, and then Innocent, even more appalled by that, attempted to annul it later in 1215, probably because he saw it as interference that would destabilize feudal/religious order.
Anyway, it was the law of the land, and as such the Church was the only real body that could print and disseminate it, so I don't think they were taking sides.