First of all, I have a dog and we sometimes walk in this area. Now to the details: there are no signs that this is a dog walking area. In google this area marked as "Imperial dog park", but I can't confirm the naming and attribution. However this area attracts a lot of local dog owners, because it is surrounded by fence and have gates. But basically the area is very huge, so you 50% of visitors - are simply laying on the ground and taking sun bath. Check this link, it has a brief video of how it is organized: https://www.change.org/p/southwark-council-keep-the-surrounding-dog-park-gates-at-geraldine-harmsworth-park