The most reliable method to find her parents would be to purchase the birth certificate - $30 - you can download a form online, but have to send details by snail mail.
Given the name was Christina Miller Smith, I looked for a marriage for aSmith and a Miller and found this
SMITH WALTER Male MILLER ISABEL FemaleBOULDER 108 1910
but it would be somewhat of a guess to say these are the parents of Christina, without more proof
Ilooked at the 1936 Electoral Roll for WA & found WA 1936 Forrest Bunbury Smith, Christina Miller, 21 Spencer street, home duties Smith, Daisy Spencer Street, South Bunbury, home duties Smith, Josephine Mary 42 Spencer Street, home duties Smith William Vincent, Spencer Street South Bunbury, labourer
which doesn't match the above names. I don't easily find a Walter and Isabel together in WA.