HEPA air purifiers are bird-safe
To clarify -- A HEPA filter in an air purifier is bird safe.
That being said, if the Air purifier (the big box with a fan that holds the filters) also includes an ionizer, it is NOT bird safe, whether it has a HEPA filter in it or not. Ozone (O3) is not great for people, and it's worse for birds . . . and even a nice fabric filter won't stop it.
I have seen some cheap Chinese Air Purifiers that include ionizers, in many cases hiding the fact they do (since O3 is a pollutant . . . it just happens to "purify" by killing things as well). UVc purifiers are bird safe, so long as the light is shrouded from the outside (who want's a sunburn?!?), but their effectiveness on air is . . . debatable.
Read carefully the purifier descriptions to tell how it "cleans" the air.
I'll echo Honeywell purifiers as good. Not cheap, mind you, but good. Picked up the "Power Plus 530" to cover the Dining room/kitchen area for our conure (Essie) and the occasional birdsitting Cockatiel visit (and the "rare" cooking incidents

). With just Essie, it was a mild improvement in air quality. Machine hums along quietly on low. When we get our visitor, it occasionally bumps up to medium, but the improvement in air quality is noticeable -- No 'tiel funk, no 'tiel fever
Also picked up a Tredy H13 sized filter off Amazon for Essie's sleeping room (small bedroom). It got knocked over a few weeks ago and wasn't working . . . took a few days for the bird funk to appear. Set the purifier back up and cleared the intake and exhausts and the next day the funk was gone again.