I believe the fee is actually $5/adult. Not sure about child pricing.
If you're a Fontenelle Forest member, it's free. Otherwise there's a drop box to pay daily admission (similar to national/state parks) $11 adults, $8 child
If you have a Fontenelle Forest membership, Neale Woods is included. If not, there is a fee. Unfortunately, I don't remember what the fee is.
Thanks! Your answer is awaiting moderation.