GEC size when adding a new meter for an ADU

[Bpscan]

House has 200amp service with a 4 AWG bare copper GEC connected to a ground rod.
We're adding a new dwelling unit in the backyard with a 125amp service. There'll be one service drop (350kcmil) connecting to a gutter where I splice to the existing meter panel (2/0 copper) and the new meter panel (#2 copper) I've already driven a second ground rod about 8 feet away from the original one. My question is, what size do all the wires connecting to the ground rods need to be? I'm looking at 250.66 and see that I need #2 for 350 amp service? Each individual meter enclosure requires smaller than #2 if I look at them individually, but I know I have to bond the two ground rods together, so is this where the #2 comes into play, in between the rods? Then a #4 up to the existing meter and a #8 up to the new meter? That doesn't seem right to me.

Thanks in advance for the help!

 

[Buck Parrish]

250.53 E #6 CU . From rod to rod to grounded conductor in the service.
Read 250.66 A

 

[infinity]

There are several ways to do this post #2 has one method that's how I would do it. Existing #4 GEC to first rod> #6 bonding jumper to second rod> #6 GEC to new service.