On the marriage cert, both Joseph and William as shown as cordwainers. This matches their occupations on the 1861 census.

William signs by a mark, so he would not have been able to read his name on the cert.

However, his father is shown as William Warren, not William Warren Parker.

There's this birth with no MMN (possible red herring):

WARREN, WILLIAM -
GRO Reference: 1844 D Quarter in NEWTON ABBOT Volume 10 Page 159