For practitioners, Parkour is a safe activity because each practitioner is entirely in control of what they do. Nobody is forced to do anything dangerous.
For authorities, Parkour is generally only a safe activity when they can be sure that the people doing the activity are responsible, and can be held responsible.
The letter as shown comes across as very emotional and the overall tone is that of a complaint not the start of serious dialogue. I don't think it would inspire the authority to trust you and your group.
Rhetorical questions ("How is it fair?") are always a bad idea. Letters of that length are a bad idea.
First impressions are important. You need to put all of the evidence that you are responsible at the front of the letter. Then, put a section on how your group benefits the University (health benefits of Parkour, involvement of students, student society etc.).
After that, you can get into the specifics of what you need. However I don't think you need to go into specifics about what your exercise involves, and you certainly don't need to do it twice. Saying that you use everything is a) wrong (would you use a fragile rooftop?), and b) counter productive (no authority will want to give you an absolutely free rein with their premises).
Instead, emphasize that although you use the University environment you only use the parts that are suitable and durable enough. Include the fact that Parkour is a long-term practice, and by it's very nature it needs to treat the environment with respect.
After that, you can start to mention how you've been training according to these principles for a long time already without causing any problems.
After that, mention that apparently someone had a problem with your group on the date in question, and that you would like to try and avoid such problems in the future by making sure that the University understands the responsible nature of your group of Parkour practitioners.