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?".
No --- they don't need to ask individual guests their sexual orientation or religion or anything --- but when a group plans on some kind of event then why not allow people to see which groups have reservations --- If I don't want my young sons standing behind the couple pictured in post 86 waiting to get on the roller coaster, then I could choose a different day.
I've made the mistake of taking my kids to an amusement park on the same day as the girl scouts --- the problem with that was the lines were extremely long and the park was very crowded --- if I had known such a big group would be there, I would have chosen another day.
But if it's being advertised as "Gay Day" by a gay group, then even without kids, if I don't want to be there on Gay Day, I should have that information.
Honestly I'd have no trouble answering any of those questions --- and maybe some --- like the second would be good to ask in a place that is meant to attract a lot of children. And like I said before, maybe certain religious groups' days --- even certain Christian groups days might make me prefer to wait for the next day --- like if you knew certain groups would be there passing out Biblical passages or trying to save you, you'd rather wait for another time.