Well, from the 1939 register Jessie was born 11th May 1885 and living at 17 Francis Street, Luton, Bedfordshire. So this should narrow down her birth registration.
Also her mother Georgina was born Scotland, and I can find a Georgina Parke b. Scotland with 5 year old daughter Jessie in 1891 at Bootle, Lancashire - but Jessie was recorded as being born Liverpool (Lancashire) ( not Birkenhead, Cheshire)
But I'm not seeing a birth registration for her.
What was her father's name & occupation on the marriage certificate?