The radio telescope pictured is at Parkes, not Perth.

There was a tracking station at Honeysuckle Creek near Canberra that did most of the work, but the Parkes radiotelescope was also used. see http://outreach.atnf.csiro.au/visiting/parkes/looselybased.html for some accuracies and inaccuracies about the film "The Dish". Honeysuckle Creek was closed down in 1981 and its dish moved to the current Deep Space Network tracking station at Tidbinbilla, also near Canberra. See http://www.honeysucklecreek.net/index.html

A tracking station at Muchea 40Km north of Perth was used for the Mercury missions. A station at Carnarvon 800KM north of Perth was used for Gemini and Apollo. Information on both of these is available on Wikipedia