If your family was Catholic, you should be able to find a baptismal record for John in Tourmakeady parish. The records only go back to 1869, but you may be able to find Mary as well. You can access the parish records at this link:
https://registers.nli.ie/parishes/1049
Civil records for Catholics began in 1864, so there are likely civil birth records for both of them (the early years can be spotty in some places, however). Civil records for births, marriages, and deaths can be searched at the Irish Genealogy site:
https://civilrecords.irishgenealogy.ie/churchrecords/civil-search.jsp
The dates may differ in the civil and parish records, because there was a fine for not reporting births promptly to the civil authorities once they started, and people often lied about the birth date in order to avoid a fine.
The baptismal record should show the townland in which the family lived at the time of the baptism. There is an online database of townlands you can search, but note that it is arranged by civil parishes (which were civil administrative divisions by then, though they had an origin as religious divisions). The civil parishes are often much smaller than the Catholic parishes, which can include parts or all of several civil parishes. If the family were actually from the town of Tourmakeady or the surrounding area, then the civil parish would be Ballygovey/Ballyovey. You can see more about that civil parish at this link, which has further links to info for each of the townlands in the civil parish:
https://www.townlands.ie/mayo/ballyovey2/
If the baptismal record shows a townland not shown at that page, then it's probably in a neighboring civil parish, which you can find through the links at that same site.
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