Hi, if the parents are James Stewart, Sarah Rodgers then there are several other births recorded on the Family search site using parent search:
23 Nov 1865 0871,COALISLAND,TYRONE,IRELAND, Female Stewart
01 May 1868 Coalisland, Tyr, Ire , Alexander Stewart
08 Nov 1870 Coalisland, Tyr, Ire , Male Stewart
11 Jul 1873 Coal Island, Tyr, Ire. , Male Stewart
15 Apr 1878 Druncran, Tyrone, Ireland , Female Stewart
These birth records also appear for Coalisland RC church on the roots Ireland site, along with the birth above from 1876 mentioned by aghadowey
Regards
Joe
PS Both parents included in both searches