I have considered this as a possibility. In the 1871 census, David and his brother William and sister Emma are listed as living with Rosina Flowerday as the head of household. It looks like David's father was either deceased very young or not in the picture. However, in other records I found a Rosina Cotterell (variant of Cockrill) born the correct year to match the Rosina in the 1871 census, leading me to believe that this may be David's mother and that there is still a male Flowerday parent. David is also the youngest of three children (all Flowerday) so I think that's a strike against him being born out of wedlock. William Cockrill may have been a grandfather or uncle. It looks like the church records from this time were destroyed in WWII, so we might never know, but I would sure like to find out!

Also, David and his brother William are listed as boarders with the Tunmore family in the 1881 census. Rosina appears to be a nurse at the Great Yarmouth Workhouse at this point -- she is listed on the staff there and the age matches up.
Edited to correct year and add info.