Hi Mike. This is John R Mullarkey III. I am 21 years old and I am from a long line of Mullarkeys from around the Salem MA area namely Peabody, MA. At this point I am only able to go back 4 generations name wise and 3 generations location wise (at least where they spent most of their lives (not sure about my grandfather's childhood, he lived in Florida for the past 20 years and only recently moved back to the area so I will finally get to know him better and will have a chance to ask him these questions.
But anyways, my father is John R Mullarkey JR. has two brothers, their father John Robert Sr. has a brother Billy (assuming William but to me he is just Uncle Billy)and there might have been others I am not sure, and their father was John Joseph Mullarkey. John Jr. and his brothers were born and raised in Peabody in the 1950s. John Senior lived in Peabody for a good 35ish years. John Joseph I don't have much information about yet but will be finding out shortly.
Just thought I would throw this out there since it seems you are (or your family was) in the area and it seems unlikely that there isn't a connection with the branches of the family being in such close proximity.