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As I see it, the Park is within their rights to prohibit him from giving out the books. The Park Authorities have the authority, as managers of the park, to set rules for its use. If the rules apply to all people equally, then there is nothing to fault. If he came on my land, I could prohibit him also. Is it a particularly good rule? No. At least not to me. It is silly, and needlessly restrictive. It is another good example of why we need to eliminate "public" parks. Privatise, privatise.
Does the 1st Amendment apply for this? It's "public" property and seems a freedom of speech issue.
"Fruitless, born a thousand times, lies barren. Unguided inspiration, yields random motion, circumscribed in destination, going nowhere. Guidance uninspired, always true in facing, stands immobile. But fixed upon that destination firmly and with inspiration lofted; beget your dreams."