Hi

RoadClosed already gave you a quick essentials overview. To answer your question,
there are too many unspecified variables.

domain names

- How do you want the two sites to be accessed?
www.mysite.com
www.myfatherssite.com

or

www.mysite.com
myfatherssite.dyndns.org[1]

Remark: dyndns.org gives you 5 free hostname entries (also available for
static IP's, what you seem to have). I guess, this is more or less what no-ip.com
offers?

- Have you already registered www.mysite.com and www.myfatherssite.com?
Usually, the registrars DO provide DNS entries as well. Enquire and insist.
If you have not registered one of the domains so far, choose a registrar who
does provide DNS entries, like domainbank.net[2], but make sure prior to
submission.

virtual hosts

As pointed out by RoadClosed, any webserver can easily configured to assign
a domain name to a "directory" (virtual hosting). This is possible since HTTP 1.1.
E.g. Apache in httpd.conf
Code:
#
# Use name-based virtual hosting.
#
NameVirtualHost 192.168.1.100

	<VirtualHost 192.168.1.100>  
#IP configuration depends on your configuration
	ServerAdmin [email protected]
	DocumentRoot "...mydirectory..."
	ServerName mysite.com
	ServerAlias mysite.com www.mysite.com
	</VirtualHost>

  
	<VirtualHost 192.168.1.100>  
#IP configuration depends on your configuration
	ServerAdmin [email protected]
	DocumentRoot "...myfathersdirectory..."
	ServerName myfatherssite.com
	ServerAlias myfatherssite.com www.myfatherssite.com
	</VirtualHost>
Cheers.


[1] http://www.dyndns.org/services/dns/statdns/
[2] http://www.domainbank.net/