Hi Caucatc,
I retired from Hong Kong a few years ago. Have a read of this working paper (number 3) in the Air Navigation Conference document. There are many papers and quite a few are available in Chinese, which may interest you. Unfortunately the working paper on parallel approaches is only available in English at the moment. This document is a few years old now but does contain a great deal of valuable information.
AN Conf/11 - Documentation
Independent parallel approaches will depend on terrain around your airport. As Frank has said you need a non-transgression zone between the approach paths, which is monitored using a PRM. If an aircraft "blunders" into the non-transgression zone then the controller is required to intervene and turn either or both aircraft. If you have terrain, as they do in Hong Kong, then this makes independent parallel approaches impossible. So you have to resort to dependent parallel approaches, which involves off-setting the parallel approaches so as not to have aircraft side by side.
So, if you don't have any terrain constraints around your approach paths, then there are no separation considerations to be made and you can run parallel approaches in CAT I, II and III, depending on your local regulations.
Good luck and a Happy New Year, On the beach
