For John Sullivan B 1860 +/- 1 yr, Co. Cork, with the father's forename unknown, I found 46 B rcds. Another 46 are likely in the mass of Co. Cork rcds which haven't been transcribed yet.
The best way to help reduce the mass of false positives is to find John's parent's names. Suggest you look for the church or civil marriage rcds for John and Helen. These are most likely to have rcd'd this info for posterity.
A second source would be John and Helen's death rcds. This assumes they both died after 1900. By 1900, most states, especially eastern ones had adopted mandatory civil B/M/D registration. The death rcds MIGHT contain their parent's names and sometimes origins.