Hi Heather,
That is the record I saw with the exception of the name not William A but William L Barry.
In addition, in the Watertown register for the 1883, there is a marriage recorded between William L Barry and Mary Agnes Smith- parents, place of birth, occupation and date are the same as in Newton. This entry is X'd out, but clearly readable.
I did not find a William L Barry birth in Watertown in 1861, sorry.
I checked for ______ Barry born in Watertown 1860-1862 and found:
Ellen Mar 20, 1860 to James and Mary (mn Barry)?
Hannah Feb 28, 1861 to Lawrence and Margaret (mn Keefe)
James Dec 30 1861 to James and Mary (mn Donelly)
Nellie Mar 27 1863 to Larry and Ellen residing in Watertown. Father occ- laborer, both born in Ireland. Sorry no maiden name- that seems to be normal for this page.
It seems like Nellie is a sibling of William L.
If you need anything else, I'm glad to help if I can.
Karen