Re: Best combination of traditions and general traits for range and damage for horse archers?
Yes, you can get horse archers with a range of 150 by recruiting Mercenary Amazonian Riders, you can see the regions where they can be recruited by scrolling down on that page.
You asked about the best combination of traditions and general traits for maximising the damage done by horse archers. You could try the advice in the thread on Maximising the Damage Potential of the Nomad Horse Archer - the game has had some patches since that advice was posted, so I do not know whether the advice still works. Maybe, when you are playing your Parthia campaign, you could post an updated guide?