His death record on the NSW BDM doesn't show parentage,
AGE 76 YEARS
but the original entry could have extra information. Assuming he was CofE, if I remember correctly, most of the historical Anglican registers have been sent to Moore College's library, so try contacting them; Catholic records may still be in the original church.
However, on freereg.com, there is only one baptism for a James Salter in Lincolnshire (seeing as he was tried in the Lincoln Assizes, he could very well be from there!)
LincolnSt Swithin Reg 206 17 June 1806
James Father: Samuel SALTER Private in the N. Lincs Militia
mOTHER Anne tOWNHILL
Hope this might be him.