Twonky is a good sign; it means that the MyCloud is running, and it has started the Twonky server, and you can communicate with the MyCloud.
There are two media serving settings; one on each Share, and then one to turn on the DLNA server; that’s under Settings|Media|Media Streaming
And, if you open up the ‘Detailed Status’ on that Twonky page, it will tell you the IP and MAC addresses of the MyCloud.
I can’t really see the first image properly.
If you connect the MyCloud to the router, can you see it using your parents’ PC?
Can you see the Twonky page (in a web browser)?
Can you see the MyCloud Dashboard?
You have read the El Capitan threads, haven’t you? And you do know there’s a bug with 10.11.1 that Apple are working on, and that the 10.11.2 beta seems to sort this problem?
And finally, you have read the user manual, haven’t you? Especially p23, about mapping the device into file systems.
If you need to set the device up, and can get to the Dashboard (the MyCloud’s control panel) on your parents’ PC, follow the ‘Mobile Devices’ setup path shown in the ‘Quick Installation Guide’ that was in the MyCloud box. That, combined with the user manual, should help you get it working.
Mac access might have to wait for Apple to sort their bug out: go and complain to them.