Hi Roderick,
The Apple Ethernet-USB adapter is made to run on your Mac. Can you establish a connection to the network on your Mac (when Parallels is not running)? If that is not the case, something's wrong with the Adapter.
You shouldn't need to connect the adapter separately within the VM as the VM uses the Mac's built-in network hard- and software to connect to the outside world. The VM uses its "virtual" network card to talk to the Mac's network system and from there accesses the outside world. The VM doesn't care whether the Mac is connected to the outside world via Wifi, or an ethernet cable plugged into it, or an Ethernet-USB adapter, or any other means. It just talks to the Mac's network system, and the Mac then takes care of figuring out what connection to use - just as it would if you weren't running the VM.
There are three ways that the VM can use the Mac's network; usually only two of them are useful (bridged, shared). Just which method works best depends on what kind of network your Mac is connected to, and if one doesn't work, just use the other. (As pointed out by YanaYana, usually "bridged" works in home networks, while "shared" may work in some corporate environments where "bridged" doesn't work).
The key point is that the VM uses your Mac's network system, and you don't actually need any dedicated drivers for the Apple Ethernet-USB connector within the VM. You would need it if you ran an OS off Bootcamp directly, but not if you run the OS from a VM.
When you connect the Ethernet-USB adapter to your Mac you should test whether the Mac is actually using and recognizing it; if so, the VM will work with it - and you won't even see the adapter in the VM as there's no need for it to show up there, since the OS running in the VM is talking to the virtual network card, and that in turn talks to whatever device the Mac is using to connect to the outside world.
You may also want to check whether the VM has "captured" the Ethernet-USB adapter by looking at the USB settings (menu: devices -> USB).
Hope this helps,
k.