I would certainly guess that the father was Gaetan/Gaetano.
I think it might be reasonable to get Ruth's death certificate to see who the informant of death was:
Ruth Mary Parrilli
born 12 December 1904, aged 92 at death
death registered in January 1997
death registered in Yeoville, 7263B
register B06, entry 157
The couple could have had children outside England and one of them might have been present at her death, for instance.
If this Count too was involved with the Italian govt during WWII, the couple may have separated and she returned to England ...
One thing just to note is that there were only ever two Ruth Mary Moulds before the 1960s, births registered last quarter 1904 (probably yours) and first quarter 1905, both of whom married in 1931. Presumably the wedding announcement you have confirms the bride's father's name as making her your Ruth?
(I imagine her father was Graham Harley Mould a barrister, although I can't track the other Ruth with parents.)
Did Ruth have siblings Graham Theodore, Ernest Harley, and Joy Gertrude Constance?
Mother was Mary Eleanor James, marriage 1898 Wandsworth, but it seems the Mould-James births post 1911 belong to a different marriage (possibly Albert Edward Mould, married 1903 Southwark), so that seems to be the total chlidren of the marriage.
Joy seems to have married twice, with 2 children to the first marriage.
She died in 2002 in Warminster.
(I will send you info about a child privately.)
Graham I find no further trace of. No wait, he died in Canada.
https://familysearch.org/pal:/MM9.1.1/J69N-F38and apparently was in the US just prior
https://familysearch.org/pal:/MM9.1.1/KY2S-YPZoddly he was a labourer ... and seems to have been 'debarred' when the ship docked ?
image:
https://familysearch.org/pal:/MM9.3.1/TH-1951-22094-63302-79?cc=1916081(father G Mould, seems to match)
Ernest on the other hand did well:
https://www.thegazette.co.uk/London/issue/42231/supplement/8901/data.pdfOrdinary Officers of the Civil Division of the Most Excellent Order of the British Empire
'Ernest Harley Mould, Esq., Principal Assistant to the Remembrancer, Corporation of London'
... but his name is too common to identify a marriage.
He died in 1986 in Chichester.
Anyway I wonder whether you could find descendants of Joy, for instance, who might know more about Ruth.