I explained in my posts above how to do what you want.
The permission error you are seeing is more than likely because you are trying to add a sub-domain to the DreamHost system when the parent domain has not been added. To prevent this you simply add the parent domain to the DreamHost system, even though you do not intend to host the parent domain here, this will not affect the parent domain elsewhere, it is strictly a work-around to allow you to add the sub-domain.
Once you have added the sub-domain, you can obtain the IP address assigned to it using the procedure outlined above and update the DNS record at your other host.
Edit: Here is a basic step by step, in case you still aren't sure what needs to be done...
1... Add 'clientname.com' to the DreamHost system using Add New Domain / Sub-Domain
2... Add 'clientarea.clientname.com' to the DreamHost system using Add New Domain / Sub-Domain
3... Obtain the IP address assigned to 'clientarea.clientname.com' by clicking the small [DNS] link for it on the Manage Domains page and examining the main A-Record.
4... Modify the DNS record at your DNS host to point the 'clientarea' A-Record to the IP address obtained in step 3.
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