Glenny/Glenney and Hazlett/Haslip, County Down - need parish information

Searching for the birth parish of James GLENNY/GLENNEY (b. 1819, Ireland; d. 20 February 1890, Ontario, Canada) and Margaret HAZLETT/HASLIP (b. 1808, County Down, Ireland; d. 17 February 1890, Ontario, Canada).  

Margaret's death certificate lists her place of birth as County Down and since she and James were married in Ireland and had at least three children there in the 1840s, we *think* that James was most likely from there as well, since there were a lot of Glennys there at that time.  James, Margaret, and their children left Ireland some time in the late 1840s/early 1850s and appeared in a census in 1852 in Ontario, Canada.

Their children are as follows:

1. Annie Maybel Glenney, b. 1841, Ireland; d. 14 January 1918, Buffalo, NY (married Robert Herran)

2. Isaac G. Glenney, b. 1843, Ireland; d. 1928, Ontario, Canada (married Ellen McCaffrey)

3. William George Glenney, b. 12 July 1845, Ireland; d. 24 January 1924, Penzance, Saskatchewan, Canada (married Emma/Emmeroy Beecher on 24 March 1874, Marshalltown, Iowa) - my great-great grandparents

4. Martha Glenney, b. about 1857, Ontario, Canada

There may be one or two other children that I do not have information about.

Since Margaret was about 33 when Annie Maybel was born (and James was 22 years old), it is possible that Margaret's maiden name is not Hazlett/Haslip and that she had been married prior to marrying James Glenney. I do not know that for certain - it is just one possibility that I thought of that could explain the lack of information we've found over the years.

Any information and leads would be most welcome!

Thank you!
Laura