Dad's computer and your computer do not need Internet access. You can setup Ethernet or other kind of LAN for sharing if physical distance is not a huge. You have to install WordPress on shared folder in that case or you an change the path of shared folder.
For the precise question "Making a locally hosted WP site viewable over the internet" - yes, definitely it is possible. All you need is a fixed IP. Localhost IP is 127.0.0.1, that is constant for all. For server we make a kind of 'alias' with a fixed IP. That is actually easy even for Windows - http://thecustomizewindows.com/2012/10/configuring-windows-7-as-web-server/
If for example, you IP is 126.96.36.199 (search with "what is my IP" in Google), pointing your browser towards that IP, usually opens your localhost for a fixed IP.
But no one surfs the web with an IP, but a domain name. So we basically need a router and set a set of NS. There are third party services which does the job for 'just testing' purposes. Some ISP provides a NS, that works fine. It is basically not that difficult, but never set such setup with your personal computer, people can hack your computer if you make a part public. A simple flaw of security can be exploited.
With Ubuntu or any Linux / Mac OS X Lion / BSB it is quite easy. Simple installation of some components turns them as a server itself.
Basically the things are dependent on a static IP and router. The Operating System / Software part is really not a major player.