Any county or city may, upon request of the governing body of any state-supported institution lying wholly or partially within the county or city, regulate the parking of motor vehicles and all other vehicles on the roads, streets, alleys, grounds and other areas within such portions of the boundaries of such institution as lie within the county or city. Any city adopting an ordinance pursuant to this section may provide in the ordinance that regulations made pursuant to this section shall be enforced by persons appointed under § 19.2-13. No penalty for the violation of any such ordinance shall exceed a fine of twenty dollars. Any request from the governing body of any such institution to the governing body of the county or city shall be in writing and signed by the presiding officers of the institution’s governing body and shall be accompanied by a certified copy of a resolution of such governing body authorizing the request to be made. The circuit court for any county or city which has adopted an ordinance pursuant to this section shall have jurisdiction to try cases arising under such ordinance to the same extent as criminal cases arising in the county or city. The provisions of this section shall not affect the application of §§ 46.2-1231 through 46.2-1234.