I did some searching for you and found proof that John Parkman's mother was indeed Anne Keverne. The first is the marriage record of John Parkman and Anne Keverne on 19 August 1779 in Falmouth, Cornwall. Here is the link:https://search.ancestry.com/cgi-bin/sse.dll?ti=0&indiv=t...
The second are the publication of marriage bans and the marriage of John Parkman and Marguerite Noel on 22 August 1810 in Verdun (attached). On the bans it gives his mother's name as Anne Keveran and on the marriage it is Keveland.
I also saw above that people wondered if they had any children in France before they moved to England. I did a search and found they had a son Jean (John) Parkman born 22 January 1812 in Verdun and died 6 September 1813 in Verdun. Both records attached.
In my search I found no document or evidence that John Parkman was called John Josiah Parkman.
Hope that's some help.
PS - I noticed that you and others don't have much information on Marguerite Noel so I did some searching. She was born (attached) on 4 March 1793 in Verdun to Nicolas Noel and Marie Françoise Petit.
Nicolas Noel and Marie Françoise Petit were married (attached) on 24 April 1892 in Verdun. Nicolas Noel was born 8 December 1866 to Jean Noel and Madelaine Alexandre. Marie Françoise Petit was born in Fleury-devant-Douamont on 8 September 1772 to Jean Nicolas Petit and Elisabeth Ferry.