I found it's armour 13 where the cut-off point is - but it still depends partially on the missile attack of the unit firing.

I did a few more tests:

It's not just armour - it's shields that count. I gave a Falxman 13 armour + gold armour upgrade = 16 armour. He got destroyed by arrows.

Then I gave the Late Legionary First Cohort a shield rating of just 1 and the archers would fire upon the unit even if it had gold armour upgrades.

Also, archers with missile weapon upgrades will fire upon more heavily armoured units. A basic archer with 7 attack will attack standard legionaries (with 12 armour), but not ones with armour bonuses. With a gold weapon upgrade (10 attack in total) the archer will target even legionaries with gold armour (15 armour). By contrast, other archer units with 10 or 11 missile rating won't attack gold armour legionaries unless they too have missile upgrades. I wonder why missile upgrades has such an effect.... perhaps the missiles have an unseen bonus that makes them more likely to pierce armour so the AI attacks armoured units with them??

I'm not convinced it's a pure bug, more of a bit of mis-calculation on CA's part. Archers are obviously made to favour certain targets, only their 'it's not worth firing at that target as I won't kill it' setting is a bit low perhaps. Worse so is how they refuse to shoot the same targets in the back when the shield isn't involved.