|There's more than one way to do things|
If the .html is indeed as regular as portrayed (and all the job specs are in a single html page as I think you have suggested), this seems almost trivial.
Based on the sample data you've shown and your (somewhat conflicted??) description of what your boss wants, I'm going to assume that you want to capture as much of the first <p> as included in the <jobname and jobserial <spans... and then skip over the (possibly outdated) incumbents, resuming your capture with the first blockquote.
What I'm hoping this labored phasing suggests is that designing a regex (or group of same) is at least as much about analysis of the source data as about coding.
In other words, it matters little whether you use a non-greedy lookahead or a negated class or something else to skip the 2nd and 3rd blockquotes (each of which happens to be immediately followed by an <a href... -- which makes them easy to distingish and thus eases the way to satisfying your "without grabbing" requirement) or any one of several other techniques that leap to mind.
Similarly, analyasis of the initial info (again, assuming regularity) tells you you want to start capturing with the line following <p><b><span class="jobname">
Following any (or, better, several!) of the approaches suggested by the above may not be what you actually had in mind, but might still serve "to expand (your) Perl skills...."
Of course, if you have text editor that will remove .html tags and supports regexen, one approach might be to simply capture the webpage source (by whatever means: save_as from a browser; LWP, etc), open the file in the editor, delete the tags and use two simple regexen to replace
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