Web parts do not follow the Parent Child paradigm. Perhaps you mean cloning.
You can clone a webpart and set all of its properties at the web part level and the use the cloned web part anywhere in the system. If you are just putting a web part on a page and setting its properties and then copying the webpart to a different page, the settings are independent although the initial copy will be identical at first. This is by design. To achieve what you want, please clone the web part, set all of the settings in the webpart properties as defaults and then use the webpart through out the system.