Since I first posted this topic I have taken a DNA test that has linked me to several other Pendletons in Canada, which will allow me to add a little more information.
I still don't know Alexander's birth place or parents, but I think I have found some of his siblings:
Daniel Pendleton, born 1789 in Drumcree, died 1850 in Ireland, married Elizabeth Farrel in 1810 in Drumcree.
Matilda Pendleton, born about 1799 in Ireland, died in Canada.
John Pendleton, born about 1802 in Ireland, died in Canada, married Ann Brownlee from Ireland in Canada.
Don't know if this helps much, but any help would be appreciated.
Gary