It could be similar. But, Parent group typically refers to a Parent Company to a subsidiary. like, Mitsubishi is the parent company, but their subsidiary is Mitsubishi Heavy industries or Mitsubishi Financial Group. In an organizational chart, Mitsubishi would be the at the Top (parent company), while the other companies would be underneath them.

Depends on the context, partent group of ____ -the group above the ____. Parent goup by itself, group for parents :) Hope this helps !