A cloud hosted database is accurate, but doesn't really describe it well. Amazon, Heroku and others also provide cloud hosted PostgreSQL, which is a very different use case.
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.
Yes, but covering all the important parts exceeds the "couple sentences / one short paragraph" goal. First and foremost it's a DB + server + client libs (I entirely agree on that one, it's a major value-prop). Details beyond that are many.
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.
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.