Parchment was made out of animal skin. The way the Egyptians would have made it is by shaving off the fur of the animal skin, then they would stretch the skin in the sun under tension. The upside of the parchment is that it preserves longer than the papyrus paper and it would not be easily torn apart because it was made animal skin.
The downside of parchment is that it would curl when there was humidity. The way they would deal with the problem is that they would use wooden book covers with straps to keep the parchment from curling.
The ancients were probably using parchment starting in 500 BC
Here is an example of high-quality parchment book:
Here is a funny skit about a monk trying to use a book for the first time: