From the North Carolina State Parks website (https://www.ncparks.gov/park-rules) "Pets are permitted in parks so long as they are on an attended leash no longer than 6 feet and under the constant control of the owner, and on all pedestrian trails. Pets are strictly prohibited from entering any building, with the exception of service animals and authorized search and rescue dogs. Owners may be asked to remove dangerous or noisy pets from the park." This includes cats.
Animals are welcome in the park. I am not sure about the conference center though.
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