Hi Ruskie
I don't know anything about your ancestors but what I can do is give some general advice on research in ireland.
Catholic Births, marriage & deaths from 1864 onwards are covered under Irish Civil registration. If you have a place and rough date (within 5 years) you can order a birth, marriage or death from the Irish GRO by post and credit card. It is cheap - costs € 6 for a photocopy of the register and only €2 if they can't find the entry.
http://www.groireland.ie/apply_for_a_cert.htmIf you don't know the place or rough date of your ancestor's post 1864 BMD, then you need to search for the event in the IGI Indexes (Latter Day Saints). These indexes are available in many countries and, with an IGI reference, you only pay €4 for your BMD at GRO Ireland.
Prior to 1864, things become more difficult from abroad and, I should imagine, much more costly.
Roman catholic parish records exist for the pre 1864 time period. Here is a website showing you which records exist for which areas. You need to click on Wexford on the map and then the appropriate area...
http://www.ireland.com/ancestor/browse/counties/rcmaps/Parish records for New Ross, for example, exist from 1789 onwards so the birth of Thomas Healy should be there.
Regarding Mary Jane Connor, if she was born in 1864 or afterwards then her birth will be covered under civil registration. If your document says she was from "Wexford, Wexford" then this means she was from Wexford Town. You couldtry ordering her birth record from the Irish GRO, specifying that she was from Wexford Town and born between 1864-1868 (these dates being the earliest 5 years in existence). It is worth a try.
The only other thing I might add that could be helpful is that there seems to have been just one Denis Conner in Wexford Town, as head of household, in 1853. The Griffith Valuation is a census substitute used by genealogists to locate heads of household. The Valuation was taken in Wexford in 1853 and is on line (although not complete)...
http://www.failteromhat.com/griffiths.phpNote, Denis Conner in the townland of Townparks, civil parish : St John's
Wexford Town. Perhpas he is Mary's father. I could not locate Darby Healy.
I hope this is helpful and good luck with your research.