Dog portraits in the park present a different set of challenges. Squirrels and other dogs that may be in the park can challenge the attention of the dog being photographed. One thing I do insist on before we go to the park is that the dog parents promise me that they are confident that their dog would not take off after a squirrel.
This being said there are benefits for taking dog portrait in a park. Certainly, the beauty of the outdoors and the possibility of many nice backgrounds. Also, natural light can often be a very flattering light.
As you can see with this portrait of a very noble looking dog the special park I use was a great place to take this portrait.