Yay, Eastbourne people!
Gildridge Park is lovely :) The cafe by the tennis courts is amazingly child friendly.
They make lovely cake on the premises, the coffee's good and they have loads of toys in the cafe and also tractors etc outside. We spent lots of time there last summer :)
Or there's the cafe by the playground too, no idea what it's like though, I've not been in there since it reopened.