Note: this is a work-in-progress and will be updated with more information over the next few days.
This guide will walk you through deploying your own instance of the open-source Parse Server. This would be a good starting point for testing your existing application to see if the functionality provided by the server is enough for your application, and to potentially plan your migration off the Parse Platform.
This guide will walk you through using Elastic Beanstalk (EB), which is an AWS service similar to Heroku. Why use EB rather than Heroku? Elastic Beanstalk does not lock you into Heroku-specific ways of doing things, is likely cheaper to run your backend on than Heroku, and it integrates with other services that AWS offer (and they offer almost everything one needs to run an application these days).