I heard about Reachability and have seen many apps using it like Facebook and Instagram, but I'm pretty sure they have their own servers and aren't using a third-party like Parse. I'm not sure about this, but doesn't Parse handle the things Reachability does?
It depends - yes on Parse, but you might want what Reachability offers anyways.
If you are ONLY making network calls to Parse, it has built-in error handlers that will let you know if it can't reach the network. So you're good there - just make sure you appropriately handle your errors and alert the user.
However, if you want something more robust, such as a toast message or a persistent message about network reachability, then implementing Reachability separately is worth it.
I have an app that only connects to Parse, but I still use reachability so I can have a cute pop-up notice that the network is unreachable. The box hides when network is restored. It was easier writing this functionality via Reachability, but you could hack a version of it via Parse's error handling if you wanted :)