Not every faction gets every ethnicity. For example as KH you get just greek ethnicities (Athenian, Spartan, Korinthian...) and no Thracians or Persians.
You can however bribe a family member from another faction that gets these ethnicies, for example a seleucid Persian or a makedonian Thracian. If that bribed guy has a son, he will have the same ethnicity as his father (but not the same name), as the trait is inherited. That way you can have your persian family as KH.