Pretty sure this is a first-hand account. The author experienced the judicial system directly, as a juror. Then he wrote this first-hand account. If you told someone else about it, that would be second-hand.

Isn't it second-hand for the parent though, which I think is what he means?

The way he used the word “vicarious” is not idiomatically correct usage.

I don’t want to put words in his mouth, but I think he might have been trying to say that the author’s description was so vivid that he could vicariously experience the jury process through reading it.

In fengwick3's comment, "vicarious" modifies "account". Assuming "account" refers to the linked article, then it's first-hand, and not vicarious. fengwick3's (or your or my) experience of the judicial system through the linked article might be vicarious, but the article itself wasn't.

Sorry for the pedantry!