It surely depends on the setting and level of power your campaign is tuned on.
In low-magic/mytological settings like Middle Earth, dragons are more intended as primeval behemots and are impervious to ALL elements, or at least near invulnerable by nonmagic cold/heat and totally immune to own element type.
In other more magical settings like say Forgotten Realms or Dragonlance, expectations on the power of dragons are more mundane. So, even if older specimens may STILL be practically immune to cold/heat AND their own element, younger more common dragons may only be immune to own element.
This is obviously IMHO only and no official ruling about it, simple the way we always handled the matter in our campaigns.