Firebase vs Parse server
Firebase is an unified and cloud based platform started in 2011 and now owned by Google which provides a set of versatile tools for building and growing mobile apps for Android and iOS devices .Firebase has many integrated services ,that makes the task of building full fledged apps a breeze, such as
A realtime NoSQL and JSON based database .
An Auth system .
Push notifications .
Admob and Analytics dashboard .
The Parse server is an open source version of the cloud based platform Parse which was shutdown recently by Facebook so the team can focus on other projects inside Facebook .To make it easy for their clients to migrate their existing apps to their own servers they have open sourced the core of the Parse platform .The project has received millions of stars and forks on GitHub and many developers have contributed to make it even better .
The parse server gives you more power and also more control since you can self host your apps with your own server but without writing so much code .You can either migrate your already existing Parse apps or even build new apps from scratch in less time without actually writing any server side code but that doesn't mean you can't ,the Parse server is extensible ,whenever you need you can extend your app with custom code and functionality .
In case you have mobile apps hosted with the Parse platform and needs a cloud hosted alternative to Parse then Firebase can be what you are looking for since it's now owned by Google which makes it a powerful and reliable cloud hosted solution for your apps otherwise you can easily migrate your apps to a self hosted server with the open source Parse server .
The open source Parse server provides you also with a NoSQL database but you need to host it by yourself .You have also automatic REST APIs and client SDKs to do CRUD operations against your self hosted database .
The Firebase database is a realtime database that's most suitable to build apps such as social networks or any app that needs real time data synchronization for all connected clients .
While Parse server can be used for building more traditional apps but also has Live queries that permit you to send queries one time and the server will take care of sending updates whenever there is new data ( which was previously queried )
Parse server uses a MongoDB for data storage and Amazon S3 static buckets to store static files .
Authentication and authorization
Both Firebase and the Parse server provide machanism and modules for adding security via authentication and authorization .They both support normal username and password auth and social based auth with Facebook ,Google etc.
The Parse server is supported by a strong open source community with millions of developers worldwide who have helped make the server even more powerful than the cloud hosted Parse platform .