I'm familiar with Android too, and via DNLA streaming was fairly straightforward with compatible devices.
On iOS the AirPlay feature can stream stuff to the Apple TV. I don't have any experience with Chromecast, so not sure there.
I have a NAS system (not attached to router) that can function as an iTunes server, streams videos, photos, etc. Don't use those features much. There's an iOS app that allows the iPhone to upload/download stuff and stream. Understand a full-blown NAS is more than what it sounds like your parents have nor need.
Some routers may have NAS like functionality and might support AirPlay as well as have an app. Hopefully someone more knowledgeable on router+hard drive streaming via iOS will chime in soon.