In-Depth Tutorial on iOS Push Notifications with Parse in 10 Minutes
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Obvious use-cases for the DB would be apps that need to share data between phones (e.g. multiplayer games/high score tables/chat etc).
You're right that it doesn't look like it's able to run any server-side logic, and if you need that you'll need a separate server, but it does provide what seems to be an easy way for that separate server to do things like send push notifications.
They have everything we need in regards to notifications. They allow us to send notifications to everyone at once, or allow users to be able to push notifications sent when they receive a message from another user.