The Mechaber in Orach Chaim Siman 470 writes that there is an opinion that holds a woman that is a first-born should fast on Taanit bechorot. The Rema adds there that this is not the minhag. The Rema however in seif bet does say that when a Father is himself a first-born and this couple has a son that is also a first-born but still a katan that the mother should fast in place of her son (since the father is already fasting because of himself). Why is it that in this case the Rema says a woman should fast for her son (when the father is unable to do so) but not for herself?
(The M”B in seif katan 9 does write that poskim say the woman doesn’t have to fast in this case as well. However I’m asking according to the Rema himself.)
The explanation is apparently that there is no rule whereby “women don’t fast”, but rather the rule is that according to the custom a woman’s first-born status does not obligate a fast.
However, where there a son who is a first-born, the son is obliged a fast – and if the father is fasting for himself, he cannot fast on behalf of his son, and the mother must fast on his behalf.