How often are (production) mobile apps using these types of BaaS providers rather than being custom-backed? Is this a tool mostly used for test/toy apps?

Good question, I'd be curious myself. I've used it quite a bit for Hackathons, since it's really easy to get started, but nothing yet in production.

One big app I know that used Parse was Band Of The Day. It was featured on the iOS App Store prominently and was ranked very high in the charts.