This is a textual issue. Most important manuscripts (including Sinaiticus, Alexandrinus, and Vaticanus) contain αὐτῶν, unambiguously 3rd masculine plural (genitive). This is the reading of the Textus Receptus that you quote However, the Vulgate contains eius, the 3rd person singular genitive pronoun that may be either masculine or feminine. This reading, as well as the one late miniscule that contains the unambiguous Greek 3rd feminine singular αὐτῆς (Codex 76), were presumably motivated by copyists’ awareness that purification laws applied to women only.
The KJV here appears to be following the sixteenth century text of Theodore Beza, who revised Erasmus’ Textus Receptus. Although the reading αὐτῆς is not found in the TR, Beza most likely elected to revise the TR by the same logic outlined above, supported by the fact that this reading is allowed by the Vulgate.
The textual situation is laid out with clarity in the NET translation notes.
The difficulty of the reading αὐτῶν is that, by Levitical law, only the mother required purification rites. (See Lev. 12:6 re. יְמֵ֣י טָהֳרָהּ | αἱ ἡμέραι καθάρσεως αὐτῆς.) According to one commentator:
It is most likely that Luke has run together the cleansing of the mother and the offering of the child...into one act....
I. Howard Marshall, The Gospel of Luke: a Commentary on the Greek Text (NIGTC). Grand Rapids: Eerdmans, 1978), pp 115-116.