Lawrence's parents' marriage was not to be found in the parish records of Attard.
A subsequent family profile (visiting all brothers and sisters of Lawrence) led to the discovery of his younger sister Anna Saveria Magdalena Paola, daughter of his parents Joanne Fenech and Maria. Her godmother was a Anna Fenech, daughter of the late Peter Fenech from Rabat.
This leads me to think that the godmother Anna is sister to Lawrence's father Joanne, making Joanne also from Rabat and possibly that's where I should be able to find his marriage to Maria.
Rabat is a town next to Attard - so this makes it very possible (as against it being a faraway town).