Hello Chris and welcome, I have done some searching on Roots Ireland to no real avail for the birth of Patrick and Bridget and their marriage, there are both Patrick and Patt Cunningham born in Louth but of the two I examined were not in Mellifont. Also saw only one Bridget Dunagan (now all Donegan I assume) but born in 1796 so too early but without parents we are in the dark anyway ans with no marriage found we have no parents.
Better news is I see a number of siblings, Richard 1837 as stated by yourself; Mary 1841; Michael 1832; John 1843 and Judith 1846. There is a gap from Michael to Richard of 5 years but I have not looked at the entries to see what they say, that is from a search of Cunninghams using the parents names. The parish records that are online are here for free but a tricky read as not searchable except by year and month. https://registers.nli.ie/parishes/0231 The area now is mostly referred to as Tullyallen and there is a church in Mell outside Drogheda.
The registers on the NLI site for Mellifont commence for marriages in 1821 so assume that is not Bridgets parish.
I live about 7 or 8 miles form Mellifont so feel free to ask questions that come to mind. There is a site for Louth here http://jbhall.clahs.ie/ or Google Louth Genealogical Sources. I see on this site a John Cunningham, freeholder in 1822 in Mellifont.
Drybridge is still there but does not appear to be an actual place name historicaly, a book locally on place names says it is where King William crossed the Boyne in 1690 and it is the site of Drybridge House. The main tourist place nearby is Oldbridge House, http://battleoftheboyne.ie/ bit further away is Mellifont Abbey ruins, http://monastic.ie/history/mellifont-cistercian-abbey/ but of course the biggest attraction for tourists in the area is Newgrange https://www.newgrange.com/ while my personal favourite and does not take a great deal of time is http://www.megalithicireland.com/High%20Cross%20Monasterboice.htm it is so serene to me, it is actually still a working graveyard and is not a long visit nor a tourist spot with other things just a small graveyard with high crosses dating from over a thousands years ago.
There is a Facebook page on Louth genealogy and another on Drogheda Down Memory Lane plus a sister page Drogheda Down Memory Lane - Lost Families but this does not appear to be very active but you could try it. There is a survey done in Louth indeed all Ireland of land owners and tenants listing only the house holder or land holder here http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch select .Cunningham and parish of Tullyallen, and Louth and there is a John in Mellifont and a Richard in Mell, done in 1854. If you have the patience you can take the site number or plot number and find it on the map icon and drag the top right hand bar to the left to see a modern version, it is tricky and annoying but can be time consuming, from these numbers you can got to the valuation office is Dublin and see how the land changed hands but it is not really a great idea unless you are really into tracing the land, as richard is the 2nd son it may be he did not get the land so emigrated.
finally of interest the main builder of covered wagons in the US was born in Drogheda, Murphy, I have a short article on him. Also if you are interested there is a 9 minute video of old photos of Drogheda showing the port were he probably left for Liverpool on You Tube called Ghosts of Drogheda.
I see your Tipperary post but will leave that for the local volunteer if there is one, however I dealt with a query there before and learned that 25 families from Tipperary went to Nova Scotia at some point and will mail the link to the man in the US you told me, think he was also looking for Carroll among others in the last few week.
Enjoy your trip,
Pat
