I can't help with his parents' marriage but his own marriage is on familysearch at "España, matrimonios, 1565-1950," database, FamilySearch(https://familysearch.org/ark:/61903/1:1:FF6T-D82 : 10 February 2018), Jayme Arbuthnot and Maria Zuazo, 07 May 1827; citing Santiago Apostol, Valladolid, Valladolid, Spain, reference 2:1GCQ72R; FHL microfilm 1,039,071.
and he does give his parents' names as "Guillermo" and "Maria Ana Arbuthnot". The film appears to have been digitised if you, or someone else, can get to a family history centre.
In addition, Aberdeen University archives has the records of the Scots College in Spain including "List of students from the opening of the College in Valladolid" 1770-1859 which might give more details if you, or someone else, can access it. The catalogue record is at http://calms.abdn.ac.uk/DServe/dserve.exe?dsqServer=Calms&dsqIni=Dserve.ini&dsqApp=Archive&dsqDb=Catalog&dsqCmd=show.tcl&dsqSearch=(RefNo==%27SCA%20CA%2F4%2F67%2F1%27)