There is a Birch baptism at Solihull 6 Apr 1697, but no first name shown, and no parents' names:
Warwickshire PR's on ancestry
These images look to be a fairly recent transcription of the early Solihull PR's or BT's - I wonder whether it is possible to view the originals? But probably the name is not legible on the original either and that is why it hasn't been copied out.