The telescope is operated by the Commonwealth Scientific and Industrial Research Organisation (CSIRO). It is one of four telescopes that make up a network of radio telescopes called the Australia Telescope National Facility (ATNF).
The Parkes radio telescope opened for research in 1961. It retains its classic look while undergoing regular upgrades to stay at the forefront of astronomical research. The telescope is utilized by astronomers from around the world.
The telescope is often referred to as "The Dish", and was the setting for a fictional film of the same name. The film imagined a light-hearted version of the real event in which the telescope was used to receive transmissions from the Apollo 11 Moon landing. There's a good article about this historic event at the CSIRO website.
For more information and interesting facts about this iconic facility, see the Parkes radio telescope home page.