SWH1, have you got a copy of the 1911 census listing for William Smith at 21 Brighton Street? It seems to me that that is the key to the whole thing. It will tell you his age and birthplace, and his marital status, and the names, ages and birthplaces of his wife and children, if any. This should lead you to his marriage certificate and hence to the names of his parents.
According to the CD-ROM transcription of the 1881 census Arthur Smith, 66, widower, painter and glazier, born Aberdeen, is at 13 Commerce Street, Aberdeen with unmarried daughter Jane, 38.
FreeCen
http://www.freecen.org.uk/cgi/search.pl lists Arthur Smith, 36, at 25 James Street with wife Helen, 20 and children Elizabeth, 14; Jean, 8; and William, 5, plus brother-in-law William Forbes, 12. Clearly, if Helen's age is correct, she cannot be the mother of the children, so it looks as if Arthur must have been married and widowed twice.
The community indexed IGI at lists the baptism on 26 June 1835 of Elizabeth Bannerman Smith, and on 29 August 1837 of Isobel Paul Smith, daughters of Arthur Smith and Martha Farquharson; and the baptism on 1 September 1814 or Arthur Smith, son of William Smith and Elizabeth Bannerman.
The 1841 census lists Arthur Smith, 26, painter, at Yeats Lane, with Isabella, 25, Elizabeth, 5 and Isabella, 3. So I now wonder if Arthur was married (and widowed)
three times, first to Martha Farquharson, then to Isabella, then to Helen Forbes?
ED2005, have you got a copy of Arthur's death certificate? This should confirm, or not, the full names of his wives.
The 1881 census lists at 3 Findlays Court, Aberdeen William Smith, 48, printer compositor, born Skene, with wife and family. This looks like the family of William Smith and Margaret Hendry, married in Aberdeen on 31 December 1859.
At 24 Rose Street is William Smith, 40, printer manager, born Aberdeen, with wife and family. This is the family of William Smith and Mary Ann Duncan, married in Aberdeen on 28 December 1864.
Both of these Williams are far too young to be sons of William Smith and Elizabeth Bannerman, whose 13 children (Elizabeth 1795; James and William 1797; Alexander 1799; John 1801; Jean 1803; Mary 1805; Thoms 1807; Isabel 1810; Hannah 1812; Arthur 1814; William 1817; Margaret 1819) were all born between 1795 and 1819, and the one born in Skene looks unlikely. However the one born in Aberdeen in 1840/1841 could be a grandson.