Well, if he read the sticky, he should have found this link already.
If you want to train at home without equipment, push-ups would be the first thing that comes to my mind. They are not that effective for gaining a lot of strength, but as long as you are "extremely weak" and can't do at least 20 of them, they serve their purpose quite well.
As you say that one pull up is difficult, this is obviously another exercise you need to work on. Don't try to avoid it, but do one pull up and try the second one... if you don't manage it, still pull as hard as you can and stay in this position for a while. At least this is how I improved from being unable to do any pull ups to doing approximately 5 a few weeks later. Just try hard enough and you will see improvements very soon.
Other great body weight exercises are Dips (especially when push-ups become too easy) and Squats. Even if you want to work mostly on upper body and abs, body weight squats (followed by pistols, once you have sufficient strength) are, in my opinion, the most important exercise for parkour beginners. You want to have strong legs to protect your joints from the typical stress of landing on hard surfaces.
I hope I could give you some useful advice, but please read all the links in the sticky as well, if you think they could be useful to you. I can only give you some limited personal experience, but no professional knowledge.