It is pretty contagious but most cases these days are mild and I believe it's passed by droplets from coughing etc... I think children are more at risk of catching it than adults due to natural immunity.
If the the father doesn't have it (I take it he has no sore throat or other symptoms?) then it's probably unlikely to be passed on that way.
I remember being dragged to the doctors with suspected scarlet fever when I was little, turned out to be an allergic reaction to Septrin - turned my face pink!