So I solved my own problem, but there are undoubtedly better ways to do this. My solution was trial and error.
First, check out an article at MacStrategy called "Virtualising Mac OS X 10.6 Snow Leopard (Server)." You can pretty much follow their step-by-step guide to installing your Leopard Server, although I sometimes preferred my customization to theirs. Either way, that gets the install done, but there's still no internet or printing.
Then, open the Server Admin. You may have chosen services while setting up the server; if not, you can do it here. The directions above tell you to call your server something like "myserver.private," which I did - and that shows up here. However, you need to click the + at the bottom; that will offer the chance to add "myserver.local," which is what you want. Now you choose services for your server to run - and here are DCHP, DNS, FTP, Print, and Software Update, among many others. I chose those because they seemed right (though I still don't know if I need both DCHP and DNS). Now click the triangle next to myserver.local, and you'll see your chosen services with little empty dots next to them. Click on one and the info panel shows that service as "Not running." But down at the bottom is the button to start it. Do that with all your services, and now you have functioning internet and print and so on (the dots have turned green).
But wait. One thing I still don't understand is, in the Parallels settings it suggests using "Shared Network." When I was having trouble getting the internet, I changed it to "Bridge / Wifi." Things work after starting the services as above, but I don't know if you need to switch to "Bridge / Wifi." If things don't work, that's the answer.
Parallels, of course, offers almost nothing useful about getting Mac OSes to run, and this is particularly glaring with these older systems, which require a Server rather than a Standard system. It would really behoove Parallels to explain how to do this.
