Your invitation to a place they cannot go on their own? You pay. You're the member, you have access. And, frankly, being able to afford a membership someplace means that you probably can scrape together an occasion fee for a friend.
What you MAY hope for is that the family reciprocates on another occasion, but you can't ask for it or expect it. If you can't pay, don't invite. It's different if you are taking your daughter to the movies and you ask if the other mom and child want to meet you there and go at the same time. That implies shared expenses.
Same thing as inviting someone to your house or to your child's party at a movie or bowling alley - your party, your expense, and you don't ask someone to pay to come to your home or your party. Sure, a birthday gift is implied, and if you invite people for dinner, someone will usually ask "What can I bring?" (at which point you decline or at least say "You really don't have to" and wait for them to insist, then assign something very small). Just as you wouldn't say, "Okay, but if you want to come, you have to underwrite half the dinner," you can't ask people to pay for your invitation or pay an "admission charge."
If the kids together decide they want to go to the lake or a public pool, then everyone can pay their share and parents on some level divvy up the carpooling and supervision. If you make the suggestion (rather than issue an invitation), you can imply the shared expenses by saying, "Hey Sue, my daughter Rachel wants to go see X movie. If your Ashley wants to go too, maybe we could find a showing they could attend together. I could drive one way and, if convenient, you could drive the other way." Same thing if they decide to go to the mall - they pay for their own purchases and you provide the transportation.
We have one son, and occasionally we took another friend of his with us on vacation - and we paid it all. We kind of spelled it out to the parents and the child, that we would cover his room, board, beach passes, meals out and family things like mini-golf. If he wanted souvenirs or a tee-shirt, or food at the concession stand beyond the picnic we packed, he was free to bring his own spending money (just as we expected our own son for pay for his personal luxuries). So as your child gets older, that sort of thing may occur more often.