I ran a 50 amp sub-panel to an outbuilding 120' from the main source panel with a four wire SOOW cable. I put the cable in an 1-1/2" conduit 2' deep. I connected the neutral to the neutral buss bar in the main and sub-panels, added a separate ground bar in the sub-panel and connected the ground wire to the main panel bus bar and sub-panel bus bar. I drove a copper rod down into the earth 8' at the sub-panel location and ran a bare #6 from the sub-panel ground bar to this grounding rod. * Is this additional sub-panel ground rod correct? *
Yes, this rod is correct, only it is NOT "additional", it is required since this feeder runs to a detached structure.
Also, as mentioned, many/most inspectors will as if the ground rod meet the 25 ohm requirement. If it does not then a second rod is required and the 25 ohm rule is no longer in place.
Also, and this is a biggie, that SOOW cord is absolutely NOT code legal in this case. Rubber cord CANNOT be used for permanent wiring. You need to use either individual THHN/THWN conductors if the conduit is complete from panel to panel, or you can use UF cable as long as the size cable you get can legally fit in the conduit. Individual conductors in conduit could be #8cu, but due to the distance and the small cost difference I'd go with #6, actually #6 for the hots, #8 for the neutral, and #10 for the ground. Uf cable would need to be 6/3.
I would seriously consider making this right and getting the whole job inspected.