I've seen videos of extreme martial artists who can do some very amazing aerial stunts from a running, standing, sitting, kneeling, or even laying down position. I have also seen parkour videos of people doing similar stunts. This begs the question, is parkour a subset of a larger system or school of martial arts? Or are the fascinating acrobatics of parkour present in many advanced forms of martial art?
Parkour is more closely related to gymnastics and acrobatics than proper martial arts (even though one of the creator of modern gymnastic was Georges Hébert a French physical educator who invented "the natural method" which consists of basic conditioning and technical moves that is still used in a few countries for the military training, this method was considered to be a martial art). That beign said, I believe classic Ninjutsu is the closest fit. Ninjutsu (or whatever the actual Ninja called it) was about more than just fighting: it was about infiltration, stealth and getting the hell out of there when the job was done. This naturally entailed being able to get into and out of places unseen, or if that fails, as quickly as possible. Oftentimes taking an unorthodox route.
I won't claim to be an expert, but my knowledge stems mostly from the books I have read on the history of feudal Japan. However, to spare you a lecture, read this: http://en.wikipedia.org/wiki/Ninjutsu#History. From the link:
Ninjutsu can also involve training in free running, disguise, escape, concealment, archery, and medicine.
Also, to vindicate myself after I was accused of being ignorant for a different but related answer, please note the following line from the link:
At some point the skills of espionage became known collectively as ninjutsu.
Parkour is a style of free running based on the philosophy of Gymnastique Naturelle de Georges Hébert. It is centered around the efficient movement from one place to another regardless of impediment.
Much of what you see is not martial arts, but a style of acrobatic performance art called XMA (eXtreme Martial Arts), spawned in gyms and dance studios.
The difference between the infiltration techniques in ninjutsu and shinobi-no-jutsu, and parkour are mostly about speed. Shinobi were not averse to taking their time. Climbing a wall was not about doing so quickly, but doing so whilst avoiding detection. On that note, to say there is similarity is to say there's similarity between bowling and baseball an account that a round object is being thrown. Efficiency in ninjutsu may be exchanged for stealth as warranted; no such thought is given in Parkour.
That said, certain techniques are incredibly similar. For instance, around 1917, Gingetsu Itoh wrote Gendaijin no Ninjutsu, of "Ninjutsu for the Modern Man". In it, he attempts to set the record straight, amidst a "ninja boom" at the time, what it meant to be ninja. Now, apparently, he may actually be the source of the term "ninjutsu", since most prior records use "shinobi-no-jutsu" or just "shinobijutsu" (I'm still slowly working my way through the text; my Japanese reading is slow). Anyhow, that said, in it, there are references to techniques for climbing and wall running very similar to Parkour. However, this was written in the early 20th century, and is not contemporary of the shinobi at their height. Therefore, usually the "big 3" (Shoninki, Bansenshukai, and Ninpiden) are considered authoritative sources for all things "ninja".
Basic acrobatics (rolling, leaping, and even cartwheels and flips) exist in many forms of martial arts. Generally, however, their use is in clearing large amounts of distance quickly, or applying total bodyweight into a technique. They are rarely flashy, and anything seen as entertaining or amazing is likely not from martial arts themselves.
Capoeira is a Brazilian martial art that combines elements of dance,acrobatics and music, and is sometimes referred to as a game.
I did this for a couple of months. There were some kicks, and a lot of acrobatics - flying kicks, back-flips, running against a wall.
What you are looking for is not a Martial Art. It is called Tricking, and it's a gymnastic sport. The goal in Tricking is to make the best, most and fastest aerial kicks, flips etc.
Wikipedia calls it a Martial Art, but it only uses moves from Martial Arts without the effectiveness. It simply uses the "cool" looking techniques several times in a row. The sport itself is not a Martial Art since it was neither meant for self defense nor combat.