Can a 45 years man that is thin and elastic learn the basic Parkour moves such as jumping over obstacles? Is it good for his body (bones, joints, cartilage, etc.)?
Taking individual background out of the situation (injuries, fears, inflexibility), yes, as in any other sport. No different than older adults doing martial arts, boxing, dance, basketball, rugby, soccer, etc.
There is risk of injury at any age. Precautions can be taken, such as improving one's flexibility and strength before engaging in a new and dynamic activity. Also good warmups and cardiovascular endurance are helpful. Starting with activities that establish the routine of healthy exercise can be practiced in stages perhaps to gauge individual needs as well. Something like starting with flexibility and movement pattern training in conjunction with building an aerobic capacity for several weeks to months, then adding strength and power, and finally adapting it all to parkour.
I do hip hop dancing as part of an injury recovery process (age 43). Nothing gets my heart rate up like that, and it trains you for precision movement (much like parkour would), and building up to difficult moves is no different than yoga or soccer in my experience. Respect the process of physiological and mental development and watch for signs of overuse, dehydration, etc., like any other sport.