React + Parse or React + Meteor?

I am trying to make a web app that let people's create website bookmarks and save their credentials for bookmarked websites, which obviously requires the app to be secure.

Since I am a novice in both Parse and Meteor, I am not sure which one to choose.

Second question, if I choose meteor, can I host my own back-end server rather than let meteor host for me?

For your second question: yes, you can host your own server.

To do so, use the following command to build and pack your web app:

meteor build {your_src_path} --architecture {os_archetecture}

This command will give you a node.js application, which can be easily deployed on your own server.

You can also build a docker image for your web app, which might provide you some extra convenience: meteorhacks/meteord