Rav Yaakov Kaminetzky, zt”l, is quoted as saying that we must bring up all of our children as if they will speak with the Shechinah. Therefore, this stringency applies to the parents of every child.
I heard in the name of Rav Leib Bakst, zt”l, whose yahrzeit was this week (14 Teves), that Moshe Rabbenu lived before Mattan Torah, at which time the stringency not to nurse from non-Jewish women was only reserved for those few who were destined to speak with the Shechinah. However, after Mattan Torah, we all speak with the Shechinah on a daily basis. Therefore, the Rema codified this stringency for all of Klal Yisrael