Hiyas,
I have never ordered asylum records from NSW State Records but viewed records/books at the National Library and most instances state parents names and descriptions. (where known)
Sometimes parent details are vague but usually a name and approx age/where abouts are supplied, when they are known of course.
((OH be warned sometimes the details can be distressing))
There are books which can have a good bit of detail and copies of various records but I could not say whether this particular family would be covered. Have a look at the National Library Catalogue and search for Asylum Randwick.
Kellie
BTW Great Job there Mary!!!