Can I run the ground wire without conduit to sub panel in the same building
In turn:


Yes, you can run the grounding wire on its own as per NEC 250.130(C) point 5.  Also, since you are using a #6 wire, it can be run unprotected as per 250.120(C), although Bare Armored Ground cable may be a better bet to keep dimwits with wire cutters at bay.  However, zip-tying the new grounding conductor to the existing feeder cable is a violation of 300.11(C).  You're better off simply fishing the wire where it needs to go, or having it take its own route to the panel.
Yes, 6 AWG copper wire is an adequate grounding conductor for a 125A subpanel -- in fact, 6 AWG copper is good all the way up to 200A.  However, if you had aluminum or copper-clad wire instead, you'd need a 4AWG wire for the job.