I live in the area and have friends that work there. Good times are before school gets out and after the kids go back. This area never really has a down time. Dollywood is closing 1/3/20 and reopens mid March. Limited operating days are also a consideration in the spring as they are not open every day. Spring arrives here in mid March with warm days and cool nights.
There are 2 parks very close to Dollywood I can suggest. The Ridge and Duvall in the Smokies. Look them up. They are nice parks. Another suggestion is when you go to Dollywood drive to Patriot Park near the Old Mill Restaurant and take the trolley from there. 3 bucks for the Trolley as opposed to 15 bucks to park at Dollywood. It takes you right to the gate too. Just remember to be prepared for heavy traffic depending on the time of year.