"Parse Server is an open source version of the Parse backend that can be deployed to any infrastructure that can run Node.js" - each of these words one-by-one do have some meanings, but altogether they have none, neither your explanation for me. I fully believe it's totally my fault and it's hopeless, so please no reply.
Parse was a:
* cloud hosted nosql database,
* with client libraries for mobile platforms,
* direct write access from mobile apps, without hosting your own API backend in the middle, and
* some methods of securing this direct write access from untrusted clients.
Other important parts may include "runs on node.js", "server code in JS", "client + server define ACL per row", "push notifications", "amazingly, supports windows mobile", etc, depending on the viewer. That's what the rest of the documentation is for, e.g. in a features list.
Just 1 paragraph about what you do is all it takes.
Look at this open source project for example: http://activeadmin.info/
The front page doesn't leave you wondering why the fuck do you want to use ActiveAdmin for.