Bagenalstown is in the Catholic parish of Dunleckney, and the Catholic parish records which are available for that parish online go back to 1820, so you ought to be able to find baptismal records there for both Patrick and Margaret, which should give the names of their parents. The records are available at this link: https://registers.nli.ie/parishes/0703
The civil parsh of Bagenalstown (an administrative district) is smaller than the Catholic parish, and you can find more info about it here:
https://www.townlands.ie/carlow/dunleckny/
As you'll see, at that site there is a list of the townlands within the civil parish, with a link to more info about each of them. If you find a baptismal record for the family in the Catholic parish register, it will often indicate the townland in which the family of the child lived, which you could then find on that list, for more info about the townland.
However, the townlands you want may be a bit further south of Bagenalstown civil parish. The family tree which you attached mentions a Kiledmund and a Ballycomack. I couldn't find townlands of those names (in Carlow or elsewhere), but on Google Maps I found a Kiledmond (slightly different spelling) about 10 miles to the southeast of Muine Bheag (the main town in Bagenalstown civil parish), and also a Ballycormick (again, a different spelling), located about halfway between that Kiledmond and Muine Bheag
Kiledmond does not appear to be a townland name (though it may have been in the past - there have been changes over time), but Ballycormick is the name of two adjoining townlands in Carlow. They are in separate civil parishes to the south of Bagenalstown civil parish (but on the way to Kiledmond, as I said). Here's a link to one of them, and it contains a link to the other (adjoining) townland of the same name:
https://www.townlands.ie/carlow/idrone-east/lorum/corries/ballycormick/
On each townland page, there is a link to the Griffith's Valuation (mid-1800's) records for the townland. I tried to check them, but could not get them to work right now. You could check them later to see whether Patrick is shown as a tenant there.
If you get a townland name from the Catholic parish register, maybe it will confirm whether one of those Ballycormick's is where your ancestors lived.
