I applaud your attempt to clarify, though I don't necessarially agree with your example. For me, the difference between parsing something and and extracting something is that when I parse something, I am interested in the whole something. When I extract something, I am only interested in a small part of the whole.
That's basically why using an HTML parser when all I want is an extraction, is so wasteful. I go to all the trouble of analysing (sometimes validating) and capturing the structure of the HTML only to throw all that effort away as soon as I have captured the bit I want.
It's a bit like carefully dismantling, labelling and boxing an entire stately home, brick by brick, when the only part of value is the Adam's fireplace in the drawing room. If the rest of the place is simply going to be discarded, there really isn't any point in doing the extra work unless some conservation or restoration is intended. In code terms, that means I am going modify, reconstuct or otherwise utilise the structure I've spent the effort capturing. In a large majority of screen-scraping tasks, the structure is simply discarded.
The argument for using a parser is that semantic knowledge gained from the structure is useful when used to locate the bits of data required, and that using the structural clues is more reliable (less brittle) when the page is updated than a dumb match by regex. The problem I have found is that when the page changes, the structure is just as likely to change in ways that require the script to be re-worked as is a match-by-context regex.
For every example one way, there is a counter example the other, and I would never critisise anyone who chooses to use a parser for this purpose. I just wish that they would give my (and others) informed and cognisant choice to do otherwise, the same respect.
"When I'm working on a problem, I never think about beauty. I think only how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong." -Richard Buckminster Fuller