Okay, with aghadowey's info and following every tidbit I could find using search engines I think I have him figured out.
1758-1769 Curate of Templemore parish, Londonderry [Derry Clergy & Parishes]
1769-1771 Rector of Termonamongan parish, Tyrone [Ibid]
25 Oct 1771-1776 Rector of Kilrea [Ulster Journal of Archaeology, vol. 2. p. 260].
20 Mar 1776-1782 Rector of Desertmartin [Ibid]
1782-1789 Prebend of Cumber Parish
1789-30 Jan 1806 (his death) Prebend of Cumber Upper Parish
At the split of Cumber parish in 1789, his son Richard became his curate in Cumber Upper parish, and his son John became rector of Cumber Lower parish. So one Richard Sr., one Richard Jr., and one John.
Thanks again aghadowey for pointing me to the Ulster Journal of Archaeology source.