It just worked.
The closest thing to "convincing" I had to do was this:
I had to tell Parallels that it should connect to the FTDI serial-to-USB adapter as opposed to just having the Mac use it. That's a normal step for hardware devices running under a VM: You have to choose which pieces of hardware that you want the host system to share with the VM running under it. And I had to install the FTDI port drivers onto the Windows 7 instance running under Parallels, but I would have had to do that even if it were a native Win7 computer.
I was able to do both of those steps without having to reboot the instance of Windows running under Parallels. After telling Parallels to connect to the FTDI serial-to-USB adapter, it immediately appeared in the Windows device manager with a yellow boo boo icon, and then I just had to go to the FTDI web site, download the driver, and tell Windows to use it.
Parallels is actually pretty amazing.