so it's not the Douglas family but the Clarke family you want info on - usual 1st step is to buy from General Registry a marriage cert which should name her father - www.manxbmd.com should be able to give you date of marriage - as before 1960 then probably worth looking at the newspapers www.imuseum.im but with one name being Douglas this would need some care in refining the search

you should be able to look at possible Elizabeth's via both manxbmd + 1901/1911 censuses - one that stands out is Elizth d/o Thomas Clarke + Elizabeth Fargher born Douglas 19000206