What are the grounding requirements for a panel fed from another panel?
The new 30A panel needs a 4-wire feeder from the 60A panel. This is two hots, a neutral and an equipment ground. 

The remote 30A panel also needs a grounding electrode (or two) since it is in a detached structure. This grounding electrode has NOTHING to do with the equipment ground run with the feeders. The two serve very different purposes.

In the existing 60A panel you will terminate the new ground to the ground bar and the new neutral to the isolated neutral bar.