Though hunting is allowed in many part of National Forest lands during the corresponding hunting seasons, I think there are definite limits on other firearms use for other purposes (e.g. target shooting) and proximity to roads, campgrounds, etc.. White Mountain National Forest- Recreational Activities
Best to check with the fish & game department in the state(s) this park is in. The links to those are also list at this Forest Service site.
Yes, you should check ahead, and get a specific citation to a specific regulation, or at least an official website address stating clearly what the rules are. Don't just take a phone call or email response saying 'yes" or "no".