One of Joseph Alexander Dickson & Lydia Ann Simpson's children married into my extended family so I have the basics but I know there are a couple excellent researchers in New Zealand which was the major destination for the Carrowcolman Dicksons and associated families.
Joseph Alexander Dickson's parents were George Dickson & Ruth Orr which is positively confirmed with his baptism at Eglish Presbyterian Church and other records. George & Ruth show in the Armagh Diocese marriage bond index in 1842.
That record showing Dickson Orr as brother in law confirms Ruth's father as William Orr and both Dickson and his sister Mary have Lisnacroy as residence at time of marriage.
George Dickson was the informant for his mother Mary's death in 1870. George had a brother Richard to went to New Zealand and I've been told records there show parents as Richard Dickson & Mary Douglas. I haven't confirmed those myself. At least 3 of George and Ruth's children went to NZ as did many other relatives and connected people.