Hello Adesso, I also can find no record of the marriage or births to those parents. There are Bollards baptised in St Mary's Parish Church of Ireland, Drogheda. This parish is in Drogheda but in the diocese of Meath but some of the parish is in Meath in a geographical sense. the C of I is Anglican.
St Marys Catholic parish has no baptismal records prior to 1834, this is the year the present RC church was opened so there may be a lack there. Marriages here are not available pre civil records for all in 1864, 1845 for Prostestant marriages.
There are some Ballards born in St Peters parish north of the river Boyne in the county of Louth and diocese of Armagh.
There are a few Bollards (I am typing name as seen) in the reocrods of St Peters C of I church, you can see them here
https://www.ireland.anglican.org/about/rcb-library/online-parish-record…;
There are no names as you gave but Bollards from 1813 I think on the baptisms.
I located two William marriages but wrong wife, one in Swords, Co. Dublin but he is in the Kings County Militia, this is now Offaly and the other in Queens County now Laois, both counties in the midlands. Marriages in 1798 and 1799, spouses names Graham and Merser.
There is a site here about Louth but little about Ballard, one poor chap sentenced to be hanged in 1793.
https://jbhall.clahs.ie/
Most church records for Dublin can be found here for free but sign in, click on church records and place Dublin, very few other church records are here. https://www.irishgenealogy.ie/en/
Feel free to ask again if you see anything that requires clarification. Drogheda was a major port at that time with a regular service to Liverpool which was a big port for migration to the USA and elsewhere. Of course there were also direct sailings to the USA from other Irish ports.
Regards
Pat