I've tried CWGC, using Geoff's Search Engine with just Parkinson and Dudley, but get nothing.
SDGW is also coming up blank.
The other possibility is that he died after being discharged, from either wounds, sickness, or non-service related (accident) and is missing from CWGC, or in the latter case, would not qualify.
There is nothing blatantly obvious on the GRO Death Index, but a couple of possibles.
Phil