If I understood you correctly, you want to give your folks a domain name that's separate from your family domain name, on which your folks can store some data. You want them to be able to access this data without getting access to your files. It's not entirely clear whether your folks want to publicise this data, or just access it for themselves and nobody else?
If they don't intend to share it with web visitors, then I'd say that Thirdeyefx is right: a htccess-secured area inside your own plan is a good way to go.
If they do want to share the data with web visitors, then I think you should:
1- register a suitable domain name,
2- have this domain hosted in your plan (you get full access to that domain, as well as the full access you already have to your plan),
3- create a new WebID for your folks (so that they get a login that's different from yours), and finally:
4- grant this new WebID access to only that new domain (they get full access to that domain but not to the rest of your plan).
Does this make sense? Let me know.