Most areas of London, apart from the oldest part - The City - are dotted with green spaces, be it smaller squares where the residents or workers can sit for a while, or larger parks. They provide a welcome respite from traffic and crowds (although some can get quite busy on sunny days) and often have additional attractions within. All of the parks are rather user friendly - there is no restrictions to sitting on the grass, and there is usually a lot of it.
Some London parks are "Royal Parks" and go back to the times when the royalty had private hunting and riding grounds protected from development. Others are results of the Victorian charitable impulses towards public improvement of the cities. Many of London's green spaces are not officially called parks, but bear names like Common or Heath - these are significantly less cultivated, less managed and not enclosed by fences or walls, but are still very much parks as the rest of the world will understand it and I am including some notable ones in this guide.
This section covers just the most iconic and most interesting of the London parks and green spaces, divided somhow arbitrarily into "central" and "outlying". Wherever you are, apart perhaps from the areas of the City and neighbouring Islington, there is likely to a be decently sized park nearby, and it doesn't take long to get to truly glorious, often ancient green spaces.
How many you visit depends entirely on the amount of time you have and the interest in park-related matters, as well as the season that your visit to London falls in. It's hard to avoid the parks in Central London, as visiting the other sights will invariably bring you near the Hyde Park, Kensigton Gardens and St James's Park. A trip to Greenwich is one of big London days out, and will naturally incorporate the park there.
Of the rest, Kew Gardens are the most unmissable attraction, but charge a hefty entrance fee. Richmond Park and Hampstead Heath are both free and both worth a special trip just for the park itself, and both not only provide grand green spaces for the visitor tired of the city, but help to understand something about London itself.