The first purchase was for 26.1080 shares at $26.0695 for a total price of $680.62. It shows ordinary income as $120.11 for that transaction so I am assuming he paid 680.62-120.11=$560.51 out of his paycheck, and the company paid $120.11 (which he doesn't claim as income until the year he sells the stock). Is that correct? It lists $743.57 as the unrealized gains for that lot.
$560.51 + $120.11 + $743.57 = $1424.19, which is the current market value of that lot.
The way it's supposed to work is that any discount on the fair market value on the date the price is established counts as ordinary income when you sell. So if the shares were worth $30 on the date the price was set and they give you a 15% discount, you pay $25.50, and the $4.50 discount counts as ordinary income. The sum of the amount you paid plus the ordinary income then counts as your basis for determining capital gains. So if you then sell for $40, you have $10 of capital gains and $4.50 of ordinary income, all realized in the year of sale.
Unfortunately IRS regulations require brokerages to report the purchase price as the basis for these securities even if they have enough information from the employer to be aware of what the real basis is. For shares acquired in 2006 this doesn't matter because brokerages don't have to report basis for shares acquired then, but you should be aware of it when reporting sales of newer shares. You will have to report a corrected basis amount on Form 8949 for these shares. See
this thread for more detail about this.
How much did your dad actually pay for the shares? Was it the full $680.62 (as implied by the "purchase price" column) or was it only $560.51? In your situation it's possible that the $743.57 is the gain calculated according to the IRS regulation mentioned above, which would actually be double-counting the ordinary income.
So either it's $680.62 purchase price + $120.11 ordinary income + $623.46 capital gain = $1,424.19 or the way you calculated it before. You'll have to look deeper into your parents' records to be sure.
The next thing that is confusing me is the dividends. On August 31, 2006 they issued a dividend of $0.22. By my calculations he should have earned 26.1080 shares x $0.22/share = $5.74. On September 22, 2006 there was a dividend reinvestment. The price on 9-22-06 was $35.35, so I calculate he should have reinvested the $5.74 into 0.1625 shares. I'm unsure why it lists 0.1118 and 0.1120 for the acquired and available for sale shares (or how those values are different. How did he acquire 0.1118 shares in 2006, but have 0.1120 available for sale? is this rounding error?).
When does he report that dividend on his taxes? Does it count as income for 2006? Or does it get to reinvest in the ESPP and not have to be reported until he sells?
You owe taxes on dividends in the year the dividend is paid, 2006 in this case. Those shares you bought with the dividend money then count as a new tax lot to keep track of, with a basis equal to the dividend amount. I can't speak to how the share amounts were calculated, you'll have to talk to the brokerage firm and look over past statements to get that answer.