Eero Tuovinen: "That whole matrix of BDTP action types has in my experience been the most confusing part of the rules for new players. My way around this is to revamp the whole explanation: instead of first explaining opposing vs. parallel actions and then going into defensive action, I do it like this:
"There are two axles of effect for any BDTP action: you have to decide whether your action will be opposing or parallel towards your opponent's action (whether your successes are deducted from each other or not), and whether your action will cause damage or bonus dice for the next round. The combination of opposing+bonus dice can only be achieved with one of the three passive skills; should you choose this special combination, you also get to change your overall goal for the conflict."
In effect, I break down the opposing vs. parallel, the defensive action, changing intent and the support action into two separate issues of whether successes are deducted and whether you're doing damage. I find that this makes it much easier to understand for the players, as they can figure out which mechanical effect components they want first, before thinking about what their characters are doing. As a bonus, this tends to school players who are keen to use the defensive and support actions; I've found that the rulebook standard way of explaining the rules relegates these options into the margins, and it takes a long while for the players to even consider them."
In the example in the Revised book, why does Gael get to Dash when he's changing his intentions?
That's an error. 
Suppose you get bloodied in the first round, then bruised in the second. In the third round, are you taking two penalty dice, or just one?
It's two, which is harsh, but them's the breaks. It should be harsh - you have enough options available to you in TSOY that you should be able to get out of most situations if you're willing to sacrifice enough. 
When characters are taking parallel actions, it could happen that both characters knock each other out at the same time. What happens then?
Clinton Nixon: I'd choose as the Story Guide whatever the most interesting option is.
John Harper: Two characters both "winning" BDTP at the same time was the best TSOY moment I've ever had.
The PC got his intention: "Kill the witch."