Help me understand the Houston park systems, especially Brays Bayou

I'd like to access Brays Bayou between Stella Link and Buffalo Speedway more easily. There are small residential streets like Lookout Court and Glen Arbor Drive that run along the park but are completely zoned no parking. Another option is to park on a residential street south of S Braeswood or north of N Braeswood and cross the big streets – not a great option during rush hours or with a pet or stroller in tow. Ideally, I suppose someone with more stamina and available time could access the park somewhere beyond that stretch and bike, run, walk to reach the green space with the pavilion.

I called 311 and asked if parking on Lookout Ct or Glen Arbor Dr could be made available to the public. I was told that the park "is for neighborhood use only" and "only belongs to the residents there", and that the City of Houston doesn't administer it. The operator thought my next option might be to call Harris County but wasnt sure. Frankly, the operator didn't seem to be able to explain who was in charge of the greenway and insisted the area "belonged to the people who live on that street only". She said that was the reason Brays Bayou did not show up as a park on her map.

So can someone explain this? Who operates the Brays Bayou greenway? Whose money was spent to build an exercise and social pavilion that only a few can access? Are there ways to get this space accessible to more people?