I have no evidence for ancient Persian arrows but it is reasonable to assume that if they were constructed in large quantities for military use the war arrows would have some sort of protection so that they are not be affected by storage or transport for long periods of time. In addition, once issued it is again reasonable to say that the arrows need some sort of protection to prevent softening from rain, fog or condensation. Thus it is likely that traditional finishes would be a local paint, varnish or wood working lacquer. In storage the arrows would probably deteriorate and so repairs might be necessary and any coating would also need to compatible with the glue used to attach arrowheads, fletchings and nocks.
Being manufactured in large numbers might mean that the factory or arsenal needs to be identifiable. Or the maker wants to advertise, or the owner/purchaser wants his arrows identifiable to discourage theft etc. So the shaft may be painted in a single colour with or without rings of different colours in a unique arrangement may well have indicated the maker or owner of the arrow.
So plain wood colour (lacquer/varnish), painted single colour or wood with coloured rings, or painted with coloured rings are all possible. Who knows? It is your army paint them as you like.