Is Smokey Bear a symbol of the national park service?

No, Smokey Bear is a trademark of the USDA Forest Service, although it is also used by other agencies for similar safety messages on occasion.

Commercial use is administered by the Chief of the Forest Service, in the US Department of Agriculture, under the rules at 36 CFR § 271.

Reproduction, manufacture or importation other than § 271 authorization is prohibited under § 261.22. Penalties may be up to $500 fine and 6 months in prison, under § 261.1b.