Is the "overlay" layer, or ROW a feature in a standalone feature class?
If so, you can use the "align-to-shape" tool to move it over (setting the Fabric as the feature to align-to-shape).
To use COTS editing/geoprocessing tools to perform this conflation, I would recommend the following tools (and these may be useful to use when simply loading parcels to the Parcel Maintenance Solution).
Manual Adjustment - tools are available to align parcels along (what are supposed to be) coincident boundaries.
Align to shape - http://resources.arcgis.com/en/help/main/10.2/index.html#//01m800000004000000
Check out Doug Morgenthaler do this against the Parcel Fabric specifically -
Automated Adjustment �?? tools are available to align parcels along (what are supposed to be) coincident boundaries.
Integration - http://resources.arcgis.com/en/help/main/10.2/index.html#//00170000002s000000
Conflation - http://resources.arcgis.com/en/help/main/10.2/index.html#//001v0000000z000000
Beza Hunegnaw - check out the MapMerger conflation tool by ESEA. They have the ability to perform an exact alignment of polygon layers. Their solution is automatic. It solves just this type of problem you're having.