From the question it is hard to establish whether the service size is 350 amps or the service conductors are 350 MCM in parallel. If the service is 350 amps, the grounding conductor is #3 bare copper wire. Parallel 350 MCM conductors will allow for a 600 amp service. The grounding conductor for a 600 amp service is a #1 bare copper wire.
It depends on the length of the run
4 sets of 750 mcm aluminum. Or four parallel runs of 600 MCM copper.
To calculate the wire size, a system voltage is needed.
500
500 mcm
It depends on the length of the run
200 amp service in chicago uses 3 aught.
4 sets of 750 mcm aluminum. Or four parallel runs of 600 MCM copper.
This size service can be wired in a few combinations. Two parallel runs of 1250 MCM, three parallel runs of 600 MCM, four parallel runs of 350 MCM. Of the three choices the four runs of 350 MCM wires would be the easiest to handle if there are any bends in the service entrance conduit.
#6 bare copper wire.
500 mcm
To calculate the wire size, a system voltage is needed.
500
250 mcm
500 mcm
six awg
Assuming this falls under the National Electrical Code (NEC) the size for the service ground is based on the conductor size of the Ungrounded (Hot) conductors. For 150A the size of XHHW Copper would be 1/0 for the Hot conductors and #6 AWG copper for the ground. Aluminum would be 3/0 for the Hot conductors and #4 AWG aluminum for the ground.