Grounding a subpanel using existing wiring
Retrofit ground. Bada-bing.

Code specifically allows you to retrofit ground in a case like this.  

Given that it's a 60A subpanel, you need to get a #10 ground wire between the subpanel's ground bar and the main panel's ground bar (which may also be its neutral bar, since neutral and ground bars are to be bonded in a main panel).  As you probably know, in the subpanel you must keep neutral and ground fastidiously separate. 

The ground wire may follow any reasonable path.  It may be legal to route to the grounding electrode system or another branch circuit's #10 ground wire instead, but I recommend going to the panel because this is a subpanel.