Our userbase was TINY (so you think scaling should not even remotely be an issue), but we had constant downtime with Parse; not to mention we observed between 1% and 3% of all requests to Parse failing for obscure reasons (and it was never clear if it was just our node, or if it was global). I seem to recall the Parse team fixing these, and then repeatedly rolling back their fixes since they would inevitably cause something even more catastrophic to happen. I am sure all of them were working really hard, and I don't want to say anything to make them feel bad, but honestly, the class of problems that plagued Parse (at least at the time I was familiar with it) did not inspire confidence. It is possible that it is much better now; I have not used it in months.
Parse is a lot of fun for quick prototypes and demos, and I recommend it for that if you find yourself wanting to iterate quickly and repeatedly on such things, particularly if most of your "innovation" is happening on a client (like an iOS or Android app, etc.).