A Greenwich pensioner is someone who served in the Royal Navy and not as a merchant seaman though he may have done both as he would have been at sea before the concept of continuous service in the navy for seamen. Because he was in receipt of a pension there will be records of John's navy service in The National Archives
http://www.nationalarchives.gov.uk/catalogue/RdLeaflet.asp?sLeafletID=41The records will not help with parentage.
The 1851 census does not confirm a birthplace of Neath.
1851 HO107 2463 folio 182
Green Neath Glamorgan
John Harris 62 St Brides, Glamorgan, Wales Head Married Sawyer Greenwich pensioner
Mary Harris 55 Ireland Wife Married
William Harris 19 Neath, Glamorgan, Wales Son Sawyer
There are three St. Bride's in Glamorgan
http://www.genuki.org.uk/big/wal/GLA/StBridesSuperEly/http://www.genuki.org.uk/big/wal/GLA/StBridesMajor/http://www.genuki.org.uk/big/wal/GLA/StBridesMinor/For the period you are interested in only St Brides Super Ely has parish register coverage on the IGI. There appears no candidate on the IGI (taken from actual parish registers as opposed to Mormon church members submissions about a birth/baptism in Neath) which would lead to the conclusion John's baptism occurred either at St. Bride's Major or St. Bride's Minor.
1841 census HO107 1421/ 23 folio 13 page 17
Green Street Neath Glamorgan
John Harris 50 born Glamorgan, Wales Sawyer
Mary Harris 45 born Ireland
Mary Harris 18 born Glamorgan, Wales
Morgan Harris 15 born Glamorgan, Wales
Ann Harris 12 born Glamorgan, Wales
William Harris 10 born Glamorgan
Jane Harris 7 born Glamorgan, Wales
Adult ages over the age of 15 usually rounded down to the nearest 5 on the 1841 census.
St Bride's major was a parish by the sea. St Bride's Minor is inland. Since he was a seaman at some point perhaps a search in St. Bride's Major parish registers might be the best starting point.
Glamorgan family History Society has produced a microfiche transcript of the parish registers.
http://www.rootsweb.com/~wlsglfhs/fiche4.htm#Church%20&%20Chapel%20Registersor perhaps some one would do a look up for you in either.
Regards
Valda