Next time you have an issue connecting with your iPad, ask what router encryption is being used? Specifically, if it's WPA2/AES? I read that some Apple mobile WiFi hardware chipsets may have occasional compatibility issues with certain router WPA2 AES protocols. Apparently, related to authentication, though haven't really followed this as I don't own any Apple mobile devices.
Since you don't have control over a Park's router encryption, there is not much you can do. However, if using your own hardware or virtual repeater to connect to the Park's WiFi, then see if changing the repeater's encryption mode to WPA2 or WPA TKIP allows you to connect?
If this is a persistent issue, then I'd take it back to Apple, and get a updated replacement iPad under warranty.