Because of the whims of history, the truly ancient part of Jerusalem has been outside its walls for about 2,000 years; the present wall is a bit shy of 500 years old - young, by the local standards. Still, when people talk about Jerusalem's Old City, those are the walls they're referring to and that's the section they've got in mind.
When Israel took control of the Old City in June 1967 there were suggestions to follow the example of Vienna or Paris and knock down the walls. A different proposal was to create a park all the way around the walls.
[source]The wasn't much room for a park along the northern section of the wall. Of course, along the eastern wall there's a Muslim cemetery, so that wasn't a possibility either. But the park could have been to the south (to the left of the picture), and of course to the west, around Jaffa Gate. This is the context of Zerach Wahrhaftig's letter to the Prime Minister of October 1, 1967 (Wahrhaftig having been Minster of Religious Affairs):
Re: The plan to create a National Park around the Old City
Since I was unable to present my full position at the recent meeting of the Ministers' Committee for Settling East Jerusalem which you chair, I am doing so in writing. I'm against the idea:
1. Unifying Jerusalem and ensuring our control of the entire city is one of our most important programs.
2. There are an estimated 25,000 people living in the Old City. The Jewish Quarter, even once it's rebuilt, won't be able to contain more than an estimated 500 families. The Jewish population inside the walls will remain a minority. In order to balance this we'll need to build right up to the walls on the outside. The plan to create a park will do the opposite; eventually, it could facilitate the creation of a Corpus Separatum in the Old City.
3. We need to attach the Old City to the modern section in a physical maner, not only a spiritual one.
4. One wall is enough. I'm not of the opinion that we should knock it down, but we certainly ought not create a second, green wall, to differentiate it from the rest of the city and give our enemies an opening to cut it off.Wahrhaftig lost that argument, and the green belt was created.
[source]Times change. These days critics of Israeli policies in Jerusalem claim that Israel uses parks to strengthen its control of the city and fend off plans for division. This blog refrains from politics, but here's a link to a page with a map of the various parks and a presentation of the argument.
Jerusalem being as spectaular as it is, I assume no one will object to another few pictures.