The offset seems too consistent to be due to the map updates that towns will do occasionally. I was thinking it was more likely that the new and old maps were re-projected from NAD83 to WGS84 differently. I thought that the old parcel map was re-projected by the osm people and hosted on their servers so MassGIS would have no idea what OSM did with their data.
I am not an expert on this stuff though. Doing some googling I found this on wikipedia:

"North American Datum 1983 and WGS 84
"The initial definition of NAD 83(1986) was intended to match GRS 80 and WGS 84, so many older publications indicate no difference. Subsequent measurements found a difference on the order of a meter over much of the United States. Each datum has undergone refinements with more accurate and later measurements.

“NAD 83 is defined to remain constant over time for points on the North American Plate, whereas WGS 84 is defined with respect to the average of stations all over the world. Thus the two systems naturally diverge over time. For much of the United States the relative rate is on the order of 1 to 2 cm per year. Hawaii and the coastal portions of central and southern California west of the San Andreas Fault are not on the North American plate, so their divergence rate differs.”
(from https://en.wikipedia.org/wiki/North_American_Datum#North_American_Datum_of_1983))

So I guess that would explain it.