In order to search the Berrick Salome parish registers themselves, I
think you would need to go to the Oxfordshire Archives.  I am assuming
you are in UK. Check before you go that they do indeed have them, and
you would probably need to make an appointment.  Details www.oxfordshire.gov.uk.
(The Berrick PRs will be part of the Oxford Diocesan Archives, which
may possibly be deposited elsewhere.)

Alternatively, you can search microfilm/microfiche copies of the same
parish registers by ordering them from your nearest Family History
Centre of the Church of Latter-day Saints (i.e Mormons) anywhere in
the world.  Details: www.familyseach.org

It is also possible to search the LDS transcripts (known as the IGI,
International Genealogical Index) online, but only baptisms and some
marriages, but not burials.  (Not all parishes have had their
registers copied.)  Again: www.familysearch.org. You need to know the
batch reference number.  For instance, if you enter 'Florence Weller'
and 'Birth/Baptism' and '1874 +/- 2 years' you will find she was
daughter of Jonathan & Mary and batch no. U000326.  If you then ask
for all Weller entries in that batch you will find five others.  Then
enter 'Jonathan Weller' 1845 +/- 10 years and you will see he was
baptised 20 May 1842 and son of Thomas and Sophie, batch no. C024291.
Do the same with that number. And so on back to Richard in 1542.  Be
aware that the IGI is not 100% reliable.  Parish Register transcripts
are usually accurate, but any submissions from individuals should be
treated as no more than a guide, to be checked by searching the actual
records.

Happy Hunting!

Finally, I am sure Malcolm would welcome details of all the family
that are not already included on his WELLER Family website.