(There was an older extension also named Parsoid which is no longer maintained or used; see Special:PermanentLink/1916875 if you want to learn about that.)
The Parsoid extension exports a REST API for the PHP port of the Parsoid wikitext parsing service. It is only required if you are running RESTBase as a caching layer in front of Parsoid. The code is hosted in the same repository as the main Parsoid library in the
extension/ subdirectory. This is a temporary hack to facilitate moving code between the Parsoid library and core.
For installation instructions, see Parsoid.