I cannot find flutter listed in your supported platforms.
You can pick your AppID and setup your Flutter project using the Parse SDK https://github.com/parse-community/Parse-SDK-Flutter. Then consume the Data :)
Flutter is also supported but we are still working on adding it to the Get Started guides. In the meantime, you can check one of the other guides and try to "translate" the code to Flutter. I will let you know as soon as we have Flutter added to the guides.
Get Started for Flutter is now available at https://www.back4app.com/database/back4app/list-of-all-continents-countries-cities/get-started/flutter/rest-api/http