Question
6 - sets of 4 600 MCM copper to service .
By 250.66
Ground wire size to Water Line?

Calculations Please

According to the table, anything larger that 1,100 kcmil requires 3/0 Copper (or 250 kcmil Aluminum).
Since you're running parallel sets, you need to run that size ground wire with each set.

Calculations are:
(6) 600 MCM = 3,600 MCM
3,600 MCM > 1,100 MCM

So with the Chart 3/0 is the Size to the water line.

A Ground wire does not go with the Conductors . (only Equipment Grounds go with conductors ,)
This is from the Service Panel to the water line.

I read the chart to say 3/0 is the biggest conductor you ever need and I agree the grounding conductor does not need to be run in each set. They are not usually even going to the same place.

The water ground is max 3/0 Cu by NEC, however the design professional may spec something larger. Terminations are at the street side of the water main, and on a compliant sized lug within the main OCP enclosure on the neutral buss.

This conductor does not go with the feeder conductors, as others have said