It's a funny thing. As this article references, every park out there has its own personality, a unique fingerprint built out of its particular amenities, its formal and informal programming, and reflective of its users and community. Yet they’re all branded the same. In some ways this makes sense. As a municipal parks department, you want visitors to know that all your properties are connected to each other and that the good things they experience in those parks are, at least partially, a result of your oversight.
But too much focus on the umbrella brand of a park system can come at the cost of the brand of individual parks. Their personalities blur into those of the entire whole. This can be a real loss since it leads to neighborhoods feeling like their park doesn't accurately reflect who they are. It's a schism of scale. While the department as a whole may strive to create and maintain a park system that reflects the city's diversity as a whole, many residents relate more readily to their neighborhood than to their city. They live their day to day in a specific place with a much narrower radius defined by transit, walkability, and their commute. In New York, the Parks Department worked to give certain parks their own personality by having a specific mobile food vendor who was particular to that park. You had the dosa guy in Washington Square, the Shake Shack in Madison Park, etc.
Whatever it is that a place is known for, own it. That's the way to build the brand and form stronger relationships with users.