Having a parent company or a holding company with one or more subsidiary companies is quite normal among (large) companies.
In English, "parent company" typically means a company with controlling ownership (51%+) of the subsidiary company. The question of control is really a separate issue, and sometimes in layman's terms "parent company" can just mean "company which owns a chunk of another company".
There will be many considerations to make, and they will be different from jurisdiction to jurisdiction. So OP should really seek the advice of a lawyer, preferably a good one who is specialized in this area of law.
How can I offer board members options only in one company and not in the actual parent company?
Just make options for shares in the subsidiary company. It really is no more complicated than that.
However in many cases a lawyer would recommend creating a shareholders' agreement in addition to the company bylaws. A lawyer who is advising the recipient of the options would probably expect such an agreement; or rightfully criticize the deal if there is no legal protection of the (minority) shareholders' rights. So we are back to "go see a lawyer". :-)
@Drew Little: I looked at your webpages, and I did not see a clear, honest business idea articulated there. I don't want to question your business without reason, and thus I'll answer your question without speculating in why you're asking the question. However, I agree with Alain Raynaud in the basic sentiment that setting up multiple companies given the business idea seems ... at least to be premature optimization, too heavy on bureaucracy and too light on business idea...
Disclaimer: I am not a lawyer, and the above should not be construed to be qualified legal advice.