I've been looking at the conditions, both in English and German (sometime things get lost in translation).
I see nothing to indicate that one parent has to have a VORTEILScard Classic to get a VORTEILScard Familie, although they do mention that a VORTEILScard Classic holder can add the VORTEILScard Familie "gratis". What I do find strange is that they indicate that with the €19,90 VORTEILScard Familie, the adult(s) also gets about a 50% reduction on tickets, which seems too good to be true, since that's no less than one gets with the €99 Classic. The difference might be that the Familie card only gets you the reduction when you are traveling with one of your children.
Austrian Rail does some strange, illogical things. For instance the Einfach Raus ticket is only valid for 2-5 people. If you are single, you can't use it. In Germany, some Länder offer both a full Ticket for 2-5 persons and a Single-Ticket for less for one person. Länder that do not offer a reduced price Single-Ticket at least allow a single person to buy the up-to-5-P Ticket. But not Austria.
And, by the way, the VORTEILScard is an Abonnement (subscription). They automatically send you a notice 2 months before the card expires and renew it if they do not hear back with four weeks that you are canceling it.
You also need a "Lichtbildausweis", a photograph ID, along with the card.