The LII article on "Publicity" says:
The right of publicity prevents the unauthorized commercial use of an individual's name, likeness, or other recognizable aspects of one's persona. It gives an individual the exclusive right to license the use of their identity for commercial promotion.
In the United States, the right of publicity is largely protected by state common or statutory law. Only about half the states have distinctly recognized a right of publicity. Of these, many do not recognize a right by that name but protect it as part of the Right of Privacy.
The Nolo page on "The Right of Publicity" says:
The right of publicity is the right to control the commercial exploitation of a person's name, image or persona. This right is traditionally associated with celebrities because the name or image of a famous person is used to sell products or services. For example, it is much easier to sell a t-shirt if there is a picture of Michael Jackson or Madonna on it. However, the unauthorized use of the image of Michael Jackson or Madonna for these purposes would infringe their right of publicity. This right only extends to commercial exploitation. Information uses such as articles at celebrity websites are permissible.
The use of the name, likeness or persona for news, information, or public interest purposes is not a violation of the right of publicity. For this reason, Taylor Swift cannot prevent the use of a photo of her in the National Enquirer. The right of publicity is not only for celebrities. Any person whose name or image is used to sell products may claim a misappropriation of this right.
The right of publicity is a matter of state law. Some states, such as California, Florida, Kentucky, Massachusetts, Nebraska, Nevada, New York, Oklahoma, Rhode Island, Tennessee, Texas, Utah, Virginia and Wisconsin have passed statutes regulating these rights. Other states do not have "right of publicity" statutes but have established common law rights under caselaw. Some states have neither a statute or caselaw regarding the right of publicity.
The Restatement of Torts, Second, § 652C says:
§ 652C Appropriation of Name or Likeness
One who appropriates to his own use or benefit the name or likeness of another is subject to liability to the other for invasion of his privacy.
a. The interest protected by the rule stated in this Section is the interest of the individual in the exclusive use of his own identity, in so far as it is represented by his name or likeness, and in so far as the use may be of benefit to him or to others. Although the protection of his personal feelings against mental distress is an important factor leading to a recognition of the rule, the right created by it is in the nature of a property right, for the exercise of which an exclusive license may be given to a third person, which will entitle the licensee to maintain an action to protect it.
b. How invaded. The common form of invasion of privacy under the rule here stated is the appropriation and use of the plaintiff's name or likeness to advertise the defendant's business or product, or for some similar commercial purpose. Apart from statute, however, the rule stated is not limited to commercial appropriation. It applies also when the defendant makes use of the plaintiff's name or likeness for his own purposes and benefit, even though the use is not a commercial one, and even though the benefit sought to be obtained is not a pecuniary one. Statutes in some states have, however, limited the liability to commercial uses of the name or likeness.
Language in the employment contract granting the park the right to use the employee's image for the purpose of promoting the park would probably prevent any suit for appropriation of the employee's image. There is no specific form or words required, anything clearly indicating that intention would probably suffice. It would be wise to have any specific language reviewed by a lawyer.
In order to file suit one would have to do research as to the exact statute or caselaw in the state involved, probably the stat where the park is located or where the employee lives. A lawyer knowledgeable in privacy law would be very helpful in such a matter. Available damages might be small, making a suit unprofitable. A lawyer could advise on probable damage amounts.
This page from Digital Media law summarizes the statutory and common-law (court made) protections for the right of publicity in Ohio, and cites several relevant count cases.
This page from Rothman's Toadmap also summarizes the state of Ohio law on this issue, and mentions Zacchini v. Scripps-Howard Broadcasting Co., 433 U.S. 562, a case in which the US Supreme Court upheld an Ohio suit against First Amendment claims (1977)