Damn you Chris

If you want to do serious parkour stop jogging and start sprinting. You'll get enough endurance by sprints. long slow running is a great way to lose muscle-mass and a bad way to lose fat
On the contrary LSD is pretty effective at fat burning, it just isn't as effective as HIIT. Do you have evidence that LSD causes muscle mass to be stripped away?
Yes you're body will eat away at muscle if you don't get enough food. That's why LSD isn't promoted for parkour. LSD burns calories real quick without using much of the energy from fat. Instead of just jogging, you'll have to do some kind of tempo or preferably interval to maintain muscle and burn fat. Gymnasts don't run a lot and sprinters don't jog for 20 minutes. Parkour is no exception.
Show me the proof that LSD strips away muscle mass. LSD in fact is very effective at fat burning. Again, simply not as effective as burning calories as HIIT.
I find your comparisons lacking. Gymnasts don't run a lot because they don't need to run a lot. Sprinters don't jog for 20 minutes because they have no purpose at becoming better at jogging. Marathoners run long distances because they want to be better at running long distances. This is obvious. If you want to get better at doing something, you do that one thing (with a little leeway), not something completely different.
You insinuate that parkour doesn't have a use for long distance running, that's not true. It really depends on what your goals are. I am an endurance athlete and traceur - and as such I've found a training method that works best to suit my needs. You want to be 220 lbs. of pure muscle? That's your decision and you can do parkour that way. I want to be 150 lbs. of super low bf% and I'll be able to do parkour through my own style tailored to my wants.
You are right on when you say to eat more. Eating more will lead to greater gains (much greater). I'm all for Mark Rippetoe and he probably knows more than me, but I'm not entirely behind eating bad foods to get greater gains. There are ways to get calories without all the negative effects of bad foods (inflammation, etc.). Eggs, milk, and olive oil for example (and steak!).