Papyrus plants grew along the Nile, the Egyptians made plenty use of it but for some reason it never really spread. They made it into papyrus (and other stuff), which is a not the same as paper. Its a lot coarser and would break when folded sharply.
Parchment and vellum are animal products and also quite expensive to make, it would never be wasted on checklists or newspapers. At the time they used wooden tablets with a layer of wax for that sort of stuff. It also doesn't come is really long rolls like the ones Spike lugs around, because its made from a sheet of animal skin that is cut into a square and dried on a rack.
The Foal Free Press episode showed a printing press (minus just about all the functional components, but oh well) with large roll of what can only be paper, using it for a throwaway news medium which you would only ever use paper for.
The ponies just use paper. Not a huge stretch, the Chinese have been making it since before the year 0. I think Twilight buys stylized paper that the ponies took to calling parchment, after the original thing that became forgotten. Also not a huge stretch; many real world words and expressions mutated in ways like that.
Or you know the show just uses whatever word sounds nice and I'm putting too much thought into this. But hey you asked