There are many places to park off-street in Hermann Park as well as along the street. The off-street parking lots are available at the large central parking lot in front of the Houston Zoo, near Miller Outdoor Theater, and at the Houston Garden Center.
A for-pay parking garage is available at the Houston Museum of Natural Science.
The Hermann Park Conservancy has a great page (click here) with a link to a very detailed map of the park.
The map is large and will take a few minutes to download if you do not have a high-speed internet connection. The thumb-nail version is below with off-street parking locations identified.