Becaue Charles is seperately the King of Australia and seperately the King of New Zealand (as well as Great Britain and various other territories), they’re not under the rule of the King of England, he just happens to *also* be the King of England.
More to the point as they are all constitutional monarchies the King is a figurehead and even his ceremonial duties are delegated to the Governor General who is effectively chosen by the Australian/NZ governments to carry out the ceremonial stuff the King would normally do. While theoretically the king/governor general has to sign every law into power and can dissolve parliament it’s a very british fudge that everyone knows he can but if he ever did parliament reserves the right to get rid of him (again).
Oddly enough Australia is the only place in recent history where the governor general screwed the whole thing up by actually using the power of the King to dissolve parliament unilaterally and caused a constitutional crisis by doing [so](https://en.wikipedia.org/wiki/1975_Australian_constitutional_crisis)
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