Plug the FlashZilla v3 device into your USB port, letting it connect to your Windows virtual machine.
Start the FlashZilla v3 software and wait for it to timeout. Then quit the program.
From the USB control menu of the Parallels Virtual Machine, disconnect the FlashZilla v3 (it may appear as 'JP6 HID' in your menu.). Make sure to keep the physical FlashZilla v3 device plugged in (USB).
Start FlashZilla v3 software again and when it starts looking for a device, go back to the Parallels USB control menu and then reconnect the FlashZilla v3.
At that point, FlashZilla v3 software should recognize your FlashZilla v3 hardware and connect successfully.