As a parchment maker, this is a question I have to contend with frequently. These terms seem to be used interchangeably, depending on who's talking. While one person will insist doggedly on one set of definitions, another will insist just as doggedly on a different and incompatible set. This is not to be confused with parchment-paper. Although, in theory, almost any animal skin can be made into parchment, sheep, calf, and goatskins are those most commonly used.
Parchment is relatively inelastic and has a firm handle when compared to leather.
The 'hair' side of the skin is that side where the hair was attached, and the 'flesh' side is the side facing the flesh of the animal. If the parchment is prepared from the full thickness of the hide, it will have a 'grain' layer on the hair side, with a distinct follicle pattern, as is the case with a full-grain leather. The flesh side will typically have a smooth, silky surface corresponding to the boundary between the skin and the underlying layers of connective tissue. In recent decades, within the trade, the term 'parchment' has often been used in reference to a lower grade product made from sheepskin splits. The skin is split into two layers by machine, and the lower layer or 'flesh-split' is made into parchment.