spect: derived from Latin "specere", meaning "look"
-ant: of doing something
So expectant means:
Of looking out for.
It is equal to "expectative", but "expectative" is neutral, rather than that "expectant" focuses more on good thing. This always indicates "a mother or father whose baby will be born soon".
Evolution of usage
The eyes of the parents of looking out for are full of happiness.
The eyes of the parents of expecting are full of happiness.
The eyes of the expectant parents are full of happiness.