1. You cannot use the same CD-Key (game will fail to start for one player). Just google "Paraworld CD Key" and you will find a new one that works. Use the Settings Tool
to erase the lines where the Key and its Hash are stored.
2. Check Windows Firewall. Maybe it is preventing it.
3. Hamachi will break Paraworlds Netcode. Make sure to uninstall it.
4. You can go to the options menu and look at its bottom to make sure that you both started the same version. Cheats will also prevent multiplayer.
5. If you play over Internet, you need to forward ports in your router. Use google to find out how you do it for your specific router. You need to forward TCP 1905 to your local machine (only the host needs to do that).
6. Check if you can ping each other with cmd.exe. If you don't know how to do that, use google
Not being able to ping the other computer usually means that you are not in a LAN (but it could also be turned off in control panel).