I confirmed in the original Townland Index book (which was used to create the SeanRuad website and my database) that the entry for Tullylish townland is correct - i.e. Co. Down, civil parish of Tullylish, barony of Lwr Iveagh, Upr Part. So I expect that the other townlands for the parish are correct also. A scan of the database shows 21 townlands in Tullylish civil parish, a few in Lurgan Poor Law Union/Registration District, most in Banbridge PLU/registration district, and all in Co. Down.
A quick look at the civil parish maps shows that Tullylish civil parish, Co. Down borders Co. Armagh - not Antrim. Makes more sense as Banbridge PLU/Registration Districts covers parts of counties Armagh and Down, and no part of Antrim - although doesn't look too far away to the north.
You can see the barony/parish and Down/Armagh county border to the on
1st Edition OSI map to the west of Tullylish. (The yellow line indicates barony & parish border, the dashed line is the county border)
p.s. given that the record mentions both Banbridge and Tullylish, I think I would say 'Antrim' on this record is a typo..