“And if her means do not suffice for a lamb, then she shall take two turtledoves, or two young pigeons, the one for a whole burnt offering, and the other for a purification for sin offering: and the priest shall make atonement for her, and she shall be clean.”
However provision was made for a lesser whole burnt offering for those who were unable to afford a lamb, a bird could be offered as a replacement (see Leviticus 1:14-17). It was this that Mary offered for Jesus (Luke 2:24), but there is reason to think that by New Testament times that had become the standard offering.
It should be noted finally that neither the woman or the child were seen as ‘unclean' in themselves. (We are not talking about sin but about ritual uncleanness). They were unclean because of the processes through which they went. But the requirement for sacrifices demonstrates that in uncleanness sin was also in mind.