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I got my set and am excited to play. I tested out the game and played a little game and this came up, I was wondering if someone could help me to see if this works.

Angharad Berightshield gets inspired when : This fighter rolls a Shield or Crit when the target of an attack.

If the triggers you flip her and then she has the ability Furious Parry Reaction: During an attack action that targets this fighter and has failed, this fighter cannot be driven back and you make this attack action. It must target the attacker.

Can she use the reaction ability right after she is inspired?

I don't have the set yet but I would say no. As it seems she inspires after the attack action. I am assuming the attack action ends if you roll a shield as no damage is done.

It happens after the attack action is done. p 13 of the rulebook:

(if the fighter is Inspired as a result of an action, the fighter becomes Inspired after that action is completed)

The way ive read this leads me to believe you do not - as the ability wasnt available to trigger when the attack was made. It says "during" which means that the parry trigger happens before the attack is over...
And by the time inspired flips, the attack is over.