Yes DNA is the only hope to find Josephs Father.
A Tree has his Parents as Joseph/Ann but that Joseph Jnr is still at home and unm in 1851.
Occp Waterman.
ANN Darke may have been buried 05 Nov 1850 Christchurch, Wribbenhall
Age 70 (birth c 1780)
1841 she is a Washerwoman
There are also a JOSEPH Dark and SUSAN Dark buried there 16 Dec 1842 and 07 Jun 1844
Age 95/90
Anns Parents?
Good ages anyhow!
Trish 
Trish. thank you for taking an interest in my post.
I could be wrong myself but I feel fairly certain that the tree you refer to has an error. It is referring to the father as Joseph Darks born in 1783. He and his wife, annoyingly also called "Ann" did have a son called Joseph Darks baptised in 1821. According to the baptism record father John was a Waterman. As you say, that father and son are both shown as Watermen living in Lax Lane in the 1851 census. In 1851 our Joseph is living with his wife Elizabeth in Severn Side.
The baptism of our Joseph clearly shows him as illegitimate and his mother as Ann. I am fairly certain that his mother Ann died in 1841 and was buried in January 1841 at Ribbesford.
Unfortunately her death does not appear in the GRO index despite the volume number and page being available on Ancestry and FindMyPast. That means that at the moment I cannot order the £3 death certificate. I have applied for her name to be added to the GRO index and all being well that might happen this week (fingers crossed). Hopefully that should mean that I can then order the £3 certificate and at least see who reported her death. Otherwise I will have to pay for the full cost certificate.
Not that any of that will get me any closer to who his father was.
You have to love genealogy :-)