There are many parks ran by different governmental entities, if you're unsure I would check the Park's website to find out who pays for it. As far as I am aware the staff might be reduced, but this doesn't mean the park will be closed. As mentioned above Burroughs Park is ran by local municipalities and shouldn't be effected by the shut down. There are many national parks in Texas that are however, like Joshua Tree National Park and Big Bend. Spring Creek Park is also open as it is a state park.
Parks are run by municipalities not the federal government.
Yes, back half of park closes at dusk.
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