I take my dog to Hermann Park a few times each month. He freaking LOVES it and is always so excited to go. It's nice to get outside, although I wouldn't go walk alone as there tend to be packs of homeless people or smoking teenagers in the more concealed parts of the trail.
The Japanese garden is especially nice. It's a great break from the hustle and bustle of Houston. Be warned that parking on a nice weekend day could take you forever. I've looked for a spot for 45 min before.
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