Hi Miss Blyth,
Welcome to Rootschat, a very useful site.
Apart from going to Gloucester and looking at all the church records for an entry for Mary Smith the only other way I can think of is to send for a birth cert or marriage cert. Even then, you cannot be sure.
There is a FreeBMD entry for Mary Smith registered 1/1921 Cirencester, the only Mary listed, which corresponds with the Gloucestershire birth Index which had only one Mary Smith with no father's name, listed 1920 Cirencester Cotswold. This would indicate illegitimacy and shows she was born 1920 but not registered until 1921.
There was no Mary Parker registered on either Index, so I would think Lucy took full responsibility for Mary, but that doesn't mean that Bertram Parker did not father Mary. It is something you may never know unless a DNA test is carried out.
A lot of things that cannot be explained fully happened in the past which makes genealogy so interesting.
Best of luck,
Elizabeth