I think Parthia counts as one of the biggest competitors during the rise period, but during the decline period they were already the replaced by the Sassanids (I'm not acquainted the internal details of how that happened).

But like Axum, neither were ever in a position where they could capitalise on the failing Roman grasp in the Northwestern Mediterranean (nowadays called "Europe"). So the pressure they applied was in the frontiers rather than the direct blows to the core of the Western Half by the Visigoths and Vandals and such.

Geography severely restricted them in a way that can't restrict China from forming economic alliance with the USA's plundering grounds, so that was the gist of what I was referring to.

But after the fall of the West, the Sassanid empire became the biggest imperial rival for Rome until the Muslim expansion made them look like rump states. So in a very contrived way, one could say Iran was always (Imperial) Rome's biggest opponent, but sadly there was no Iran in Britain.