Yes. Thomas Austin was an english migrant, and in the early days the english were responsible for the introduction of a number of "vermin" in an attempt to replicate their social lifestyle from the old country, Foxes, rabbits, and various fish including the trout were all introduced, and now considered in many circles as vermin. Rabbits to the the extent that large scale disease programs were introduced to get rid of them, and trout to the extent that they have now put the existence of many species of native Australian fish in great jeopardy.
The upshot is, "thank your mother for the rabbits" is a disparaging remark, similar to "thanks for nothing", and reflects the attitude of a lot of Australians about the early english basically stuffing up the country.