I grew up in Pocatello in the 1950s (until the 70s) and I don't remember an Orphanage Rd. Do you know other streets close to it? My youngest son lives and works near Old Town.
The river that ran through town is the Portneuf. I know of a Catholic church and LDS church ward that were in that area and one of the hospitals was called St. Anthony's.
Hope that helps. I'll check my resources and see if they know of an orphanage.