We are getting too detailed here. Regardless of what it ever meant, there are two basic assumptions made which remove all issues.
One: The early Romans were Edomim. Therefore, the entire Christian empire which came from that (after Rome conquered most of the world and the Roman emperor Constantine converted to Christianity) can legitimately be referred to as Edom.
Two: The early Moslems were Yishma’elim. Therefore we can assume that the entire Muslim empire that rose from there can be referred to as Yishmael.
What to do with Paras I admit is not entirely clear. I believe that they are close enough to Yishma’elim that Chazal would not have had a problem with combining the two for some purposes (and certainly nowadays when Yishma’el controls the lands that were then controlled by Paras). I cannot say that I have any strong enough proof for this though, so I really cannot respond if you disagree. I don’t think I’m wrong either though. Both seem fairly valid.