There are three points in the story where Zack thinks his father has died. The sewage treatment plant in the Back Story, destroying the Disruptor in Antarctica, and finally destroying Admiral Vance's control center in the climax. It isn't until the third death that he is Killed Off for Real. Xavier Lightman used up all three of his lives, in true gamer fashion.
In a more metaphorical sense, Admiral Vance is now on his third life. His first life ended when he was recruited into the EDA, his second life when Zack and Xavier foiled his misguided attempt to wipe out the Europans. His third life started after that, having realized the depth of his mistakes and willing to admit to his error.
While Vance and Xavier are pretty obvious foils, with Vance being the straight-laced soldier and Xavier being the gamer who has issues with authority, Vance and Zack also make for a pretty interesting comparison. Personality differences aside (Zack is definitely his father's son, after all), they both have one important thing in common: They both make serious mistakes due to an incomplete understanding of what is going on.
Zack chases the Glaive into the hangar bay, not realizing that the hangar bay doors are designed to slam shut unless they pick up a friendly transponder signal: Zack's interceptor, hot on the Glaive's tail, essentially holds the door for the enemy to get in just ahead of him.
Vance thinks he is facing a fight to the death with aliens from the Jovian moon of Europa. What is actually happening is that humanity (under his leadership) are being tested to see if they have the capacity to set aside their fight-or-flight instinct and take part in interstellar society. Vance nearly made a costly mistake because he wouldn't listen to Xavier and Zack, just as Zack made a costly mistake because he wouldn't listen to Vance.
