Okay. I am somewhat confused as to what you have in the way of service or features, but I will try and explain it anyway. Forgive me if I get into the obvious. I do not know your knowledge level. I will also tell you I do not know cPanel. I configure things manually and have a Virtualmin set up that I can reference but that does not help you. As well, I have to make some assumptions due to lack of information. I will work from these assumptions.
When you have a hosting service, it is either for a single domain or set up for virtual domains. I am assuming that you are set up for virtual domains. Virtual domains allow you to host more than one domain name on a server, virtual server, or some other mechanism. It is my understanding is that control panels default to a virtual domain set-up.
There are two things that need to happen; setting up the DNS and setting up the server. You have two places you may need to look for some of this. It may not all be in one basket. I will get into that.
On the DNS, each domain name must have an IP address associated with it. Example.com would have an IP address 10.0.0.100 using an A record. An A record associates a name to an IP address. There would either be a CNAME or A record for www making www.example.com reach the same location. CNAME is an alias and does not point to an IP address but a domain name that already has an IP address. The alternative would be to assign www to the same IP address as the domain name, however, using CNAME is often best. Unfortunately, sone DNS servers and control panels are adverse to this and will require an A record instead of a CNAME. IF this is the case, then go for it. Just do not be surprised.
I cannot tell you that setting up a DNS record in cPanel is the answer for you. I am assuming that is is. I do not know if you are hosting your domain names with your registrar(GoDaddy) or web host. You can try setting up appropriate DNS entries using cPanel and see what happens. Otherwise, you will have to use GoDaddy which I assume is where your domain names may be DNS'ed. If this is the case, then logging onto your GoDaddy account will allow you to first check to see if your previous domain name is DNS'ed with GoDaddy then add another if necessary.
I am assuming Apache is your web server. Most installs are LAMP installs which means Linux, Apache, MySQL, and PHP (was PERL historically).
On the server side, you would need to configure your web server to handle incoming requests for the new domain name. I am assuming Apache, but you may have another web server. In the control panel, you will be adding a new web site. On some configurations, a parked domain is a completely different set-up and will not work if you want to create a new website. Generally speaking, most of the defaults will be fine, however, you will need to specify the domain name and likely the document root. Some configurations will fill in document root for you. If this is the case, then likely this will be correct according to a templating system.
According to http://www.siteground.com/tutorials/cpanel/addon_domains.htm, it looks like you will be adding an Addon domain which may include other features such as FTP, e-mail, database, and so on. The documentation I found starts here: http://www.siteground.com/tutorials/cpanel/
Since I do not know cPanel, I will give you some guidance that I am hoping to indicate that you chose the right options.
In /etc/apache2/sites-available or /etc/local/apache2/sites-available, you will see some files with a naming convention like domainname.tld.conf. You will have one per domain name that Apache hosts. If you do not see these file names, then it may be that your configuration is set up for a single domain using the file default. This would be somewhat rare using a control panel. If you set up your new domain name correctly, you see a file for this domain name.
You will want to edit the configuration file to know where the document root is. Take note of this and then cd to that directory and make sure there is an index.html file or maybe a index.php file. Edit this file and take note of the content. Then test your site again to know if you are visiting the site correctly.
I am hoping that this is enough to get you where you need to go. If not, add a comment and I will try and fill in the holes.
