I can't help with the ID (nice plant, nice flowers, not one I recognize), but I can shed some light on the question about names. The genus Parodia absorbed a few other genera when the related cacti (which were distinguished primarily based on different seed types, as I understand it) were determined to be too similar to be separate. Among those genera are Notocactus, Eriocactus, and others. So what used to be Notocactus became Parodia, and you're right, Karen.
This was as of a few years ago. I don't know what may have happened in the intervening years. My reference is dated 2001 and there is a comment to the effect that if two additional related genera are also reduced to synonymy (Blossfeldia and Frailea), the whole apple cart has to be renamed Frailea because that name is older than Parodia.
