There isn't really an equivalent naming convention at the demand stage. You could call them demandor/demandee, but it would sound ridiculous.
If you need generic terms, you would probably use names that describe the nature of the relationship: husband/wife, landlord/tenant, debtor/creditor, buyer/seller, etc.
In my experience, though, good demand letters, like any good legal writing, refer to everyone by name. People frequently identify parties by their role in the case, but mostly just because that's how everyone has done it for a long time, and lawyers are terrified of doing anything new, especially in writing. Courts and law schools are generally pushing people away from this convention, but many people are still hanging on.