I've transferred my parse.com app onto a Heroku App with MongoDB - https://github.com/ParsePlatform/parse-server-example (Heroku seems to be the most popular way of migrating away from the discontinuing service).
I can communicate with the REST API & write using the application key set in the Heroku setup. I now want to lock down API access, so that only logged-in users can call the API (set up as database stored users - not oAuth via Twitter, Facebook, etc).
This would have been done by the following in Parse.com (in app) App Settings > Users > Allow anonymous users