The places that you mention are off limits to both CC and OC (regarding alcohol, admission, etc). A bowling alley doesn't have an admission charge - they have usage charges - for the lane and the shoes (at least that is how I see it). Most bowling alleys I have been to serve beer, though, so you may be SOL just for that. Skating rinks I believe are the same way - pay for usage of the skates and the rink but I'm not sure. Otherwise, these places are private property so their own rules would apply.
As for parks, you can OC or CC in national parks. As for state parks, you can only CC. You can CC in local parks as well, but watch out for some of the local ordinances that restrict usage in certain limited areas (and some localities are actually defying state law by attempting to restrict more than what is allowed in the state law).
If you are on NC state gamelands, you can OC or CC as long as you are not hunting with the weapon. There are some privately owned gamelands that do not allow OC/CC - go to ncwildlife.org for more info. You can CC at boating and fishing access areas, but no OC.
You can also OC or CC at state owned rest stops along the highway.
Disclaimer: I believe the above to be correct but IANAL.