I replayed the story mode recently and noticed something new. When the Dragonborn learns a shout he/she also learns the understanding of who taught them the shout. Before the Dragonborn learned the Dragonrend shout he/she was warned that by learning this shout, the Dragonborn would also accept the understanding of what this shout meant.

The Dragonrend shout was made by men out of complete hate and despise for dragons. And made during a time in which dragons tormented and massacred men. Once learning this shout the Dragonborn accepted this hate and inability to forgive Parth for all of the unspeakable atrocities that he committed to the men that created the understanding of this shout.

I believe that Arngeir warned the Dragonborn and didn't want him/her to become corrupted. People like to compare Parth with a high ranking Nazi general; imagine if this reformed Nazi general opened up an organization to help the families of concentration camp victims. It would be easy for us to think "huh.. maybe this person deserves a second chance to repent for his crimes", however what if you were a concentration camp victim during WW2 and held that understanding of deep rooted hate and despise; you would want that mothersensor to rot in hell.

Think about that for a little bit.

Just because the DB accepts the understanding of why Dragonrend was created doesn't mean it becomes him. He still has control over his actions and has his own beliefs.

So is killing Paarthurnax the right thing to do? Or is it the right thing to overcome hatred, learn forgiveness, and give him a second chance?

The thing is that YOU would forgive Parth, but it would be impossible for us to understand how our Dragonborn understands this shout. It would be like being a cop and arresting a criminal for murder, than compare that to being a cop and arresting a criminal for murdering your wife and family. Changes things a little bit right? The Dragonborn accepted this intense hatred into themselves, to the point that perhaps forgiveness is not an option after that point.

That still doesn't mean the DB is unable to forgive Parth.

Your own example, it would be like a cop arresting a criminal for murdering his wife, then finding out the guy was forced to do it when he was 10 years old and is now 80 (Never mind the math here) or else he would have been killed himself. There is a level of understanding there. Parth was forced to do what he did, and while the motives of those of the past are valid and understandable, this does not make it justification to kill Parth.

Even the most intense of hatreds can be overcome. Forgiveness is always an option, and vengeance is not a good thing.

I can't bring myself to kill Paarth. He did evil things but that was thousands of years ago... during the Merethic Era... it's now the 4th Era... the fact that he hasn't done squat but teach The Way of The Voice tells me he isn't a threat anymore.

You're misunderstanding the meaning of Dragonrend.

The Dragonrend shout was created to essentially give dragons the understanding of what it means to be a mortal for a few seconds, something that they can't usually understand. It was, however, created BY people who lived under unimaginable cruelty, as WEAPON against the cruelty of their masters.

As Arengir explains, understanding a thu'um is to take the inborn meaning of it into you, NOT the reason of it's creation. The words for Dragonrend are Mortal, Finite, and Temporary.

Can you explain how those words mean "unforgiveness to dragons, they must all be exterminated." And then explain why the Dismay shout (Fear, Run, Terror) or Marked for Death (Kill, Leech, Suffer) aren't WORSE for the Dragonborn to learn, turning them into an evil tyrant?

To take a thu'um into you is to incorporate it as part of you. That doesn't mean it dictates your actions, just gives you a deeper meaning into certain concepts than what you normally would have.

As someone that typically kills Paarthurnax anyway, I can honestly say I haven't looked at the issue from this angle before.

I normally just ask myself, "What would Stannis do?" Hasn't failed me yet.