As arguments between churches go, this one isn’t the worst I’ve seen.
See the rest of the story here.
If you want want to know my view on who is right; the answer is neither of them, though the Catholics are doing better than the Presbetyrians. The mistake both churches are making is to think that heaven is the place where we will spend eternity rather than on a renewed earth. Of course a renewed earth will have dogs on it (and trees, and a full ecosystem). Whether our own dogs will be ressurected to the new earth, I very much doubt – but I do expect that the new earth will have perfect (house trained) dogs.