I'm sorry but i've been visiting this park for ages and you never had to pay before. Let me tell you, the Botany Bay National park is TINY. Especially in comparistant to other National Parks where you don't have to pay for entry. I think its like $6.50 for entry.
Unique Suggestions: Make most of it. Do all the walks you can, take a look at the mueseum.
Fun Alternatives: You can always walk through Captain Cooks landing place, and then walk up from the bush to the National Park. It's quiet a walk and they may fine you if you have no ticket anyway.
There are a few bushwalking tracks that people miss. The walks are only 1-2km long but theyre pleasant.
The Views are quiet fantastic, no matter what weather you come in, you will always see things differently at Botany Bay.
Fondest memory: The deep blue sea.