"Books existed in Egypt long before they came into use in Greece," according to the Oxford Classical Dictionary (OCD, 3rd ed. revised). "Systems of writing had been invented and developed for administrative purposes in both Egypt and Mesopotamia by c. 3000 BC."
Vellum is a type of parchment made from animal skins. From what I can tell, the widespread use of parchment was a relatively late development. Most early books were written on papyrus. According to the OCD, "parchment played only a minor role compared with papyrus, which remained the dominant writing material throughout Greek and Roman antiquity," but "leather was also used in Egypt to produce rolls for literary texts." According to Herodotus, "the Ionians also used leather rolls at a time when papyrus was scarce."
Note: If, like me, you can't understand half of what this girl is saying, closed captions are available.
Vellum was considered to be the "best quality of prepared animal skin," according to Wikipedia, so it was generally reserved for important literary works, religious texts, and the like. At the Library of Alexandria, papyrus rolls were sometimes kept "in a vellum cover with a colored label," according to the OCD, and "parchment codices had come into use for classical literature by the 1st cent. AD." Papyrus and parchment sheets were sometimes mixed together, perhaps to save money.
Parchment gradually replaced papyrus as the centuries went by. "For codices, the advantages of parchment over papyrus are obvious: less fragile folds, greater durability, greater capacity, and they are easier to use. Constantine ordered 50 parchment copies of the Scriptures for the churches in Constantinople, and Jerome records that the papyrus manuscripts in the library of Caesarea, having become worn by use, were replaced by parchment codices." (OCD)
For important books, at least, it sounds like parchment replaced papyrus in the same way that DVDs replaced videocassettes. This is a classic example of technological innovation in the ancient world.