RCC says making a woman a priest is as sinful as abusing a child.

[Thor]

Um, no, local parishes don't fall under the responsibility of the Vatican

Baloney. The RCC has a hierarchy, does it not? The local parish falls under the archdiocese, which reports up the chain. The Vatican is at the top of that chain.

--and they only fall under their authority in the sense of the responsibilities of Faith+Morals.

Authority comes with responsibility. Can't have one without the other.

I am under the authority of the Vatican and the president as well, in the sense that they both can influence laws which I follow.

The difference here is that you are not compelled to follow the dictates of the Vatican.

In terms of the minutiae of running a local parish, there is no way that the Vatican could run all the parishes in the world.

No kidding! Just like the President could not possibly run all the agencies of the US government. This is why people are appointed at various levels to handle day to day chores. But the President is ultimately responsible for those agencies.