As they died in the UK and CWGC site says "investigating..." my first port of call would be the 1911 census and then in a large town or city old maps particularly useful are:
http://maps.nls.uk/geo/explore, the National Library of Scotland have the whole UK covered and interactive
First see if the nearest cemetery has records online, e.g. (gravestone photographic resource, familysearch) for free at first and if you are local then visit, war graves are well-known plain white rectangular headstones. (I am more saying this for those unfamiliar with the subject, not you). Any marriage records easily found for the parents should narrow down where the parents married and that churchyard or its more recent burials site would be a good port of call, you may be willing to make a discreet enquiry of the church as to what churchyard was used in that timeframe.
Finally, without turning over too much ground, the newspapers and institutional records should bring the graves/service to light with a following wind, but in my opinion a lower % chance and a lot to sift through.
