As Larry suggested, the "expert" solution was to edit /etc/network/interfaces and change:
#iface eth0 inet dhcpto:
iface eth0 inet dhcpi.e. uncomment it.
As he also suggested, there is a way to do this from the GUI. When I went to System > Administration > Network I saw this:
[Notice the title bar says "Network Settings" although the menu option was just "Network". I always give my programmers heck for that kind of inconsistency.]
"Roaming mode" ??? I selected Wired Connection, clicked on properties, and changed it to:
[Yet more inconsistencies - I selected "Wired Connection" but I got "eth0".]
i.e. un-checked roaming mode and picked DHCP.
This has a similar effect to the "expert" method, adding a line to /etc/network/interface:
iface eth0 inet dhcpI can see roaming mode might be a good choice for laptops, but it seems odd that it installed this way. Maybe something in Parallels makes Ubuntu think it doesn't have a regular wired connection. It would be nice if the network icon options at the top of the screen included an option to "save" your choice of wired networking (or just did it automatically).
Now when I reboot I still have a network connection. The tooltip on the network icon now says "Manual network configuration" which doesn't seem quite right to me - DHCP is pretty automatic. But I guess it's more "manual" than "roaming mode" (whatever that is).
I feel a little stupid at not having sorted this out myself right from the start but you can't win 'em all, I guess. Thanks Larry!