Comparing the letters with neighbouring words, could the last part be hill? An entry, two lines above, looks like Hillman. Also, the h’s on this image and a neighbouring image looks similar to the (possible) fourth letter from the end of the word for William Cooper’s entry. So, possibly B_____hill?
I believe William married in Shipley, Sussex (I need to double-check my family tree). I don’t believe I’ve ever found proof of his birth. [Years ago, I did post topics about the Coopers and Parkers but I don’t believe I’ve researched them lately. Certainly not for this problem.]