Table 250.66 calls out your mandatory grounding electrode conductor.

For the purpose of using that chart, if you have 4/0 hot conductors with 2 parallels, 4/0 = 212 kcmil. So two of them is 424 kcmil.

Edited to clarify.

 

The conductor to a ground rod never needs to be larger than a #6 copper.

 

I have #6 installed, but when my inspector was here for my rough-in, he noticed I only had a 4/0-4/0-2/0 AL going from the outdoor 200A disconnect thru the wall into the indoor 200A breaker panel and he told me I will need a #4 copper for ground between the boxes.

Why would I need a larger ground further downstream than the actual earth ground?

Note:
I have a 400AMP Meter base that feeds into two outdoor panels with thru lugs. the outdoor 200amp panels have 200AMP breakers that serve as the disconnects. One enters the barn/apartment building and the other is for a future detached house, so will only have a single 20AMP outdoor outlet for now.

EDIT: While I was typing this, C'est Moi posted what seems to be ana answer to this question.

 

The terminology is grounding electrode conductor for the conductor going to the rods. Normally a 400 amp service would need 1/0 or possible #2 copper for the grounding electrode conductor but the code states there is no reason to be larger than #6 to the rods, as Jim posted. I think the reason is that an 8' rod in the soil will do no better than the equivalent of a #6 wire so they gave us a break. lol

 

This sounds like a GEC vs EGC thing. IE, the problem from colloquially calling too many things called ground & ground wire.

GEC = goes down to the ground rod.
EGC = most other "ground" thing. Definitely applies to your box to box thing.

EGC needs to be bigger since it has the job of clearing faults, so needs to carry enough current to trip the breaker.

GEC does not carry fault current in that sense.

Sometimes you there are things called grounds that are combination of GEC/EGC so you need to take the strictest size between the two, used to come up in solar circuits.

 

The EGC needs to clear faults. The GEC is for high voltage events like lightning and do not clear faults.