MsJamie

For those who stumble across this while searching, this is what I ended up doing, after much trial and error.  (Much error.)
 
1. Disabled parity.
2. Attach the 3 new data drives to the array (format, etc.)
3. Created a new share (imaginatively named "clone") and set it exclusively to the new drives.  I didn't need to lock out the new drives for the existing shares, as nothing was being written to the server during this time.
4. Open a root terminal window
5. rsync -av /mnt/user/sharename /mnt/user/clone/sharename
6. Repeat step 5 for each share.
7. Profit!
 
Yes, I'm a CLI nerd.  The advantage of using rsync from the command line is that I could interrupt it at any time with a Ctrl-C (I'm in a truck; no spinning drives while the truck is rolling, please), and re-entering the command will pick up where it left off after a few seconds.