Whilst following another discussion, some things occurred to me to say, because I see that there are some misconceptions about parish registers floating around.
1) If there are some parish registers that exist from, oh, let's say, around 1617 -- since that was the document in contention that I noticed -- that does NOT mean that those are all the parish registers that there were, and so therefore all the babies that were born at that time are in those records. No, no. Alas. There are so many lost documents that those of us who have spent decades finding things and transcribing them are Made Sad. Even though the bishops were later taxed with copying everything, that doesn't mean that they had access to everything in their bishopric that had ever existed, OR that their copies exist. Alas.
2) it is not true that mothers were not usually listed in the birth records. That changed from parish to parish, and it changed from time to time, and it changed from clerk to clerk. Here, for instance, is a page from Bristol, 1607 -- https://www.geni.com/documents/view?doc_id=6000000177873751855 -- they regularly gave the mother's name in baptismal information.
3) it is not true that the occupation of the fathers was never given. It was not given as often as the information about mothers, but it was given when the occupation mattered, either because it was an important occupation in itself, or because it was a locally important occupation, or because there were fathers with the same name and the parish was used to telling them apart by mentioning their occupations. Here, for instance, is a page from Warwickshire, 1616 -- on May 17, one of the entries reads "Zachari Gibson sonne of M[aste]r Samuel Gibson vicare," in this case an instance where the occupation was important in general.
4) Transcriptions are great! Yay! Glad they exist! HOWEVER. If there is access to either the original document, or a photograph of that document, the document transcends any transcriptions in importance. This is because it is absolutely possible for transcribers (even me!) to make mistakes. And it's possible for those mistakes to be handed down, and even original documents misread, for decades. In the 40 years of Bassingbourn churchwardens accounts that I transcribed back in the long ago, for instance, I found a mistake in the transcription that had not only been handed down, but upon which entire theories had been based. For decades. This is true even for the bishops' transcriptions. No transcription is a document. For that matter, no photograph is the document it's a photo of. There is very very very often information on the actual paper or parchment page that is not picked up in a photograph. However. To see the documents one has to travel to where they are kept. So, yay! Photographs! Yay!