though I'm sure training in the day(s) you're fasting wouldn't be pleasant...
Actually, exercise while fasted is usually great, at least for me. Remember that, as Chris Said, those who IF (intermittent fasting) tend to have 18ish hour fasts, so you aren't going an entire day without food. Instead, you space out your eating times and just eat more within those times. You stack your eating into a meal and space it out between the next big meal.
Although, I have to mention (before I forget to type it) that parkour training can sometimes last for hours. You may take a break, walk to a new spot, then train again....and so on. While training most parkour movements, you're probably doing mainly power and plyometric work. If you're fasted and you train like this, put some food in there no later than an hour or so after training.
If you don't want to eat a whole meal before continuing to train (resonably

), then I'd suggest easily portable fatty foods (basically nuts and seeds).
My reasoning is, if you're already in a fasted state, exercise will likely feel great (relative, though). But, parkour training varies from traditional exercise. If you have no responsibilities for the day, then it's common to train for most of the day, with of course the aforementioned break in between.
However, this means if you hold your fast for the entire time you parkour train, you can likely start pushing the fast into a detrimental zone; with the power work mixed with general endurance (walking from spot to spot and the like).
To my knowledge (and please, Chris/Steve/Charlie correct me if I'm wrong...I don't want to be wrong and continue to misinform other people This may be pushing your reserves (especially! if you're a skinny person to begin with (like myself)) and become a problem.
But other than that, typical exercise and in general, fasting (I have definitely found) is a great dietary tool.