Han Solo & the 12-parsec Kessel Run

Yeah, he's definitely boasting, and possibly exaggerating. If you want to take the track that Han is just blowing hot air and totally claiming something ridiculous because he thinks he's dealing with a couple desert rubes (I think that was mentioned above), that makes some sense. The character reactions in the movie would fit that, too.

What we don't have is any real evidence that Lucas screwed up and misused parsec. Believe me, I wish that was the case, because the guy's a douchebag that is better off dead. And I'd love another instance to pile onto him about being an idiot. However, that's not the case here. Sadly.

 
I always thought Ben shaking his head at Han was because the joke was that the Falcon made the run in a measurement of distance instead of a measurement of time.

Like me joking about how I ran the Boston marathon in 24 miles.

 
The distance is essential to the plot point that the Falcon is fast. We're all on board with Speed = Distance/Time, I presume. Basic law of physics. The topography of that stretch of space is the key - it's a field of black holes, the edges of which the pilot has to approach with enough speed to achieve escape velocity from the gravitational pull of the black hole.

It's the old penny-in-the-cone thing. Looks like this:

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The penny goes down the chute into the cone, rolls around and around and then falls through the hole in the bottom. Penny = ship, cone = black hole. The chute isn't long enough, doesn't give the penny enough speed, to achieve escape velocity. Give that penny a 20-foot long chute and it rims out because it has enough speed, it has reached escape velocity.

Or, think of escape velocity along the lines of putting.

See Brandt Snedeker. Brandt is the captain of a slow ship. Brandt flies the Kessel Run on the Falcon's path. Brandt does not have enough speed to reach escape velocity. Brandt is sad. Brandt is squished.

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Bye, Brandt.

See Tiger Woods. Tiger flies the Millennium Falcon through the Kessel Run. Tiger's ship is fast. It has escape velocity. Go Tiger, go!

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See Tiger go. Tiger is happy! Tiger is not squished!

@Mavric - Lucas definitely used parsec as a measure of distance, not time. With the explanation above, it (hopefully) makes more sense why.

Wookieepedia has another take on this, from Lucas:

In the commentary for Star Wars: Episode IV A New Hope DVD, George Lucas mentions that the parsecs are due to the Millennium Falcon's advanced navigational computer rather than its engines, so the navicomputer would calculate much faster routes than other ships could.
Clearly, Lucas is a moron. And before anyone says, "Well, George Lucas said it, so it's canon/true/heretical to disagree," this is the same George Lucas who brought us Jar Jar Binks and edited Star Wars to make Greedo shoot first, so he can go suck eggs.

 
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