Since Dollywood is a privately owned place, it's really within their rights to allow (or not) any article of clothing they want. It's not much different than a club having a no-jeans policy, or a store refusing service to someone not wearing shoes.
But those policies usually have a reason behind them. The park spokesman said Dollywood requires clothing or tattoos that could offend others to be covered. Shouldn't that policy be better written? A t-shirt supporting the Boston Red Sox could be potentially offensive to New York Yankee fans in the park - so would a Red Sox fan be forced to remove his t-shirt?
That's extreme, of course, but the point is that any message or idea has the potential to offend someone. An arbitrary policy like this just seems too open to interpretation to be really effective. Presumably Dollywood is an equal opportunity employer that doesn't discriminate against sexual orientation. So if the park employee taking tickets that day had been a gay man or woman, would the t-shirt have been thought potentially offensive? Probably not.
Years ago, I was in Disney with X. He chose a particularly graphic t-shirt to wear when we visited MGM Studios. It was blatantly offensive - but he was admitted to the park. Later on, he was asked to reverse the shirt because others had complained. He didn't mind - he actually felt bad because he wasn't even thinking about the message on the shirt when he got dressed that morning (frankly, neither was I). It didn't offend either one of us, but it did others, and since they paid for the same tickets, and were upset enough to complain, it had to be addressed.
Doesn't a policy that says you may be asked to cover clothing or tattoos if other patrons are offended make sense? If the point is to "protect" others, shouldn't they be the ones to decide what they do, and do not, find offensive? Otherwise, isn't it really just an arbitrary rule that allows the personal preferences of park employees to be enforced?