I reached out to St Patrick in Franklin County. They did not become a parish until 1866. I have reached out to Venango Historical Society. Incredibly helpful and thorough in records searches for me. Didn't find a birth record but did find at least tax records of supposed father.
St. Patrick's did recommend me to contact the Diocese of Pittsburgh as Irwin Twp borders that diocese. St. Anthony in Forestville did not open until 1907. The diocese of Pittsburgh told me the only parish in 1810-1811 was St. Patrick's in Pittsburgh. So my guess is the children were baptized by traveling priests.