I'll begin by saying that in 5th ed. there is no "default setting" outside of the Multiverse. Published adventures for 5e currently take place in Forgotten Realms (with the exception of Curse of Strahd, which is in Barovia, which lies within the Domains of Dread, a demiplane that can be dropped into any setting), but it has not been designated as the default.
That being said, yes, paper exists. Check the chapter on equipment and you will see in the table titled Adventuring Gear, amongst the many items:
\begin{array}{l l|l|l} \text{Item} & & \text{Cost} & \text{Weight} \\ \hline \text{Paper} & \text{(one sheet)} & 2\text{ sp} & — \\ \text{Parchment} & \text{(one sheet)} & 1\text{ sp} & — \\ \end{array}
Item | Cost | Weight |
---|---|---|
Paper (one sheet) | 2sp | — |
Parchment (one sheet) | 1sp | — |
There is also a "Book" item, separate from the "Spellbook" item, which is described thusly:
Book: A book might contain poetry, historical accounts, information pertaining to a particular field of lore, diagrams and notes on gnomish contraptions, or just about anything else that can be represented using text or pictures. A book of spells is a spellbook (described later in this section).
There is nothing here to indicate what materials a book might be made of, but it is probably safe to assume that books could be made with parchment, paper, or vellum pages.