When I try and add "domain.com" under "Addon Domains" I get this error
Error from park wrapper: Using nameservers with the following IPs: 38.97.225.165,4.79.81.160,38.97.225.164,4.79.81.159 Sorry, the domain is already pointed to an IP address that does not appear to use DNS servers associated with this server. Please transfer the domain to this servers nameservers or have your administrator add one of its nameservers to /etc/ips.remotedns and make the proper A entries on that remote nameserver.

The domain is pointing to 4 Name.com servers.

its because you have not changed the nameserver to the ones that is with the server. Once you have done this and waited for it to change threw out the internet which takes 24-72 hours. and then try to add it and it will let you.

Or ask your hosting provider to add it, he can do it before the domain is pointing to your server.

So I can't use the Name.com servers? I use them for my primary domain and just use an A record to point to the server IP

If you want to use Name.Com managed DNS, you can do A record to point to the server IP, it will work.

I did, I have domain1.com and domain2.com both using an A record pointing to the server IP. domain1.com works, but domain2.com just shows the "You have installed cPanel" page. And when I try to add it as an addon domain, it gives me a DNS error.

Edit: Should have mentioned it's a shared host. I know with a VPS you can edit the apache file and make a virtual host or something. Can you do this with cPanel?

You'll need to contact your host and explain your situation. They should be able to accommodate you one way or another.