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.
Hooboy. This may be prohibitive to bring to full code, so let me discuss some options to get the most safety increase for the least buck.
Romex NM-B is not legal for direct-burial or for running in conduit. But what really has me spooked is bootlegging a neutral off that ground wire. If anything breaks in that ground wire, it will energize every ground in your electrical system at 120VAC - touching a conduit, the panel, light switch screws, all the things that are supposed to be safe will be dangerous! Far away from the house you could be dead before they find you! Do not leave this situation to fester merely because bringing it all the way to current code would be hard: Don't let "perfect" be the enemy of the "good". Every option here restores the proper function of the safety ground and separates neutral from it.
Ed Beal discusses an issue with the ground rod.
I am willing to assume some defects may be grandfathered or permitted via local exemption.
Option 1: Go 120V-only
If you can bear having only 120V, then remove all markings from the white wire. In the main panel, black goes to a 120V breaker (or one side of a 240V breaker), white to neutral bar, and bare to ground bar (they may be the same in the main panel). In the sub-panel, black wire goes to one phase, white to the neutral bar, bare to the ground bar which are isolated. Every other row of breakers will not work - just move your needed breakers to rows that do work.
Cost: $30ish per building, for a separate neutral bar for the sub-panel
Option 2: Go 240V-only - temporary transformers for 120V
If your primary loads are 240V-only and/or multi-voltage loads (lighting, some gadgets), and your 120V loads can be hooked up temporarily as needed, easy - just reconfigure for 240V-only. Stop using neutral; the bare wire is ground-only.
Mark the white wire with tape on both ends to designate it a "hot". In the main panel, black and remarked-white to a 240V breaker. Bare to the ground bus. In the sub-panel, black and remarked-white to opposite "hots", bare to the ground bar. No neutral bar - no neutrals will exist in this panel! No exceptions! Every white wire in wiring will be marked with tape as it's not a neutral.
To power 240V loads, done. Many machines will run on 120-240V. Lighting particularly - check your fluorescent lights, the newest ballasts are 120-277V automatic switching, if yours isn't, just change ballast, they're $10-20. Good time to upgrade to T8 bulbs while you're at it. Many LED bulbs are 120-277V, I've been replacing 175W barn lights with 15W LED lights that are $10 and multivoltage. Old mercury, sodium and halide lights have transformers, and they are often jumpable for 240V. For plug-connected devices that can accept 240V, change your wired receptacles and plugs to NEMA 6-15 or 6-20.
To temporarily power loads that are 120V, use a common step-up/step-down transformer.
Cost: est. $250 per building for new bulbs, ballasts, outlets, plugs, power supplies and a step-up/down transformer.
Option 3: Go 240V, install a 120V transformer permanently
This is like Option 2, except with a permanent 120V transformer feeding a third panel.
I often find 5KVA transformers around "used" for about $100. These have 240V on the primary, and 20 amps of 120/240V split-phase on the secondary (jumpable for 40 amps of 120V only). Do the sub-panel per option 2, and add a 20A breaker to feed this transformer. The third panel gets its hots and neutral from the secondary of the transformer. It gets its ground from the ground rods feeding the sub-panel. Because the third panel is fed from a transformer, it's actually not a sub-panel at all - it's a main panel, and you do bond neutral to ground here. The transformer makes this safe.
You only have 20A per leg (or 40A at 120V-only) - so keep the loads here to a minimum.
Cost: est. $300 per building for used transformer, enclosure, additional mini-panel, breakers and wiring.
Option 4: Use a transformer for all loads.
Here, we use a larger transformer (12 KVA or larger, single phase) to feed the entire sub-panel. Like above, this makes it a main panel.
In the house's main panel, we mark the white wire with tape, and punch it down into a 240V breaker. Ground goes to the ground bar. At the transformer, we jumper the transformer's primary for 240V and attach the black and marked-white wire. Ground goes to the transformer chassis or enclosure.
We jumper the transformer's secondary for 120/240 split-phase. Its hots and neutral go to the 2 hots and neutral bar. Being fed by a transformer makes it a main panel, so ground and neutral are bonded together and the ground must go to the rods and also to the transformer chassis. The now-main panel is hooked up in the normal way. This gives you best of all worlds at higher cost.
Cost: est. $500 per building for a used transformer and wiring; $1000 for a new one.
Option 5: Pull 4 wires (in existing conduit??)
ThreePhaseEel addresses this well in his answer. You're pushing the limits of what that conduit can handle, out and in, and it's going to be one miserable pull. I prefer to install conduit much larger than the wires need, mainly to make the pull easier (and provide room for expansion). I would bring in an electrician for this - simply because they'll have all the right pulling tools on the truck, and the skill to do the pull without tearing up the wire. The right tools make all the difference in the world.
Keep in mind if there's any damage to this conduit, the pull will fail and you will end up digging up the yard. Water in the conduit is no big deal; that's why you use THWN wire.
And don't presume it's in conduit. If it's direct-burial cable, Code requires the cable exit the ground in conduit. That would explain using barely-big-enough conduit that he couldn't possibly pull 200 feet through. If it's direct-buried, you'll have to re-trench it.
Cost: est. $700-1200 per building: $400 for wire; $300 for tools or $800 to hire an electrician for the pull. If the conduit is not pullable, more.
Best Answer
You absolutely, positively cannot do that. Period.
Both hots and the neutral must all come from the same panel and one, only one, 2-pole breaker.
If you think you have a balancing problem, you could move some circuits around - but honestly, you'll want to get some help with that. Seriously.