They lived on the Isle of wight Hampshire, Tinkers lane when they returned to England. His Father was John Parkman and he married Ann Keverne in 1799, Falmouth. I don't know any more back beyond that date until I can confirm exactly who her parents were. several Ann Keverne's were born within a similar timespan, all in St Keverne, Cornwall - See Cannon Diggers transcripts on the St Keverne web site.
Do you have Ann Parkman's brothers & sisters? I take it this is Ann Parkman born circa 1827. Have you tried the Isle of Wight Family history society site as they have online BMD index from 1837 onwards - searchable by mothers maiden name & father's surname. The 1841 & 1851 census list the family details.