Since the HOA owns the shared alley the road can be tagged with operator=(the HOA name). Since the HOA owns the service road that means any resident can use the road.  

BTW  - I live in a HOA with some HOA owned roads. Those roads are open to the public, not just the homeowners.  In your case, if a delivery is to be made to one of the houses serviced by the private road, the delivery driver can use it. So can guests. 

I would recommend adding the operator tag and removing the access tag since there is not signage restricting use.