Why not warn people --- even as someone suggested if certain religious groups were having a meeting there that day --- maybe you don't want to be handed the Watchtower if it's Jehovah Witness Day --- but you could decide for yourself and your own family. It's not just Gay Day --- but what if you'd rather not bring your family on Sadists day or whatever else?
First of all, it's not in any 'profit-seeking' companies best interests to track this sort of thing. They want "everyone's" money.
If the Islamics descend upon your park (assuming they are minding themselves, no protests, no chanting, ect.), and your sign says "xxxxx welcomes Allah Akbar's University", you have just associated yourself with this group. Furthermore, you may lose sales to people like us, who may see this sign and decide to go somewhere else.
The parks are intended to provide amusement, rides and fun. That is all; and that's all they want. It's basic capitolism - maximize profits. Would you want the park to say "Today we are x% white, y% black, z% hispanic, and w% Asian, religions break down into these categories a% Christian, b% Athiestic, c% Wiccan, d% Buddist, etc, etc."
How would you respond if you were asked at the gate the following questions:
Are you homosexual?
Are you sexually attracted to young children?
What religion best describes your background?
Did you bring your domestic partner?
and so on, and so forth.
These lines of questions are personal, and are frankly an invasion of your rights to privacy. The only question a park has the right to ask is "How to you intend to pay for your ticket?".