Guns by airplane; no problem but remember:
1. You have to tell the person at the ticket counter you are traveling with sporting arms (you could also say guns or rifles but sporting arms is airline lingo and sounds better to them).
2. Guns have to be in a locked hard sided case. You take the gun from the ticket counter to TSA and they inspect the case before your eyes and lock it.
3. Ammunition has to be in some kind of container meant for ammunition. Factory boxed ammo is fine as is ammo in plastic ammo boxes. Some airlines want the ammo in a different piece of luggage than the guns are in and others don't care. You are limited to 11 lbs. of ammo. There will probably be an oversized baggage charge on the rifle case.
4. DO NOT try to take a handgun into Canada. If you're stopping in Canada ONLY travel with long guns. It's no problem transiting Canada with long guns but I don't know the current rules since it's been a long time since I hunted in Canada.
5. I'm quite sure there are hunting seasons in AK that are also in times of good fishing. Some friends and I hunted deer and ducks and fished for halibut and salmon last October in Kodiak, AK. I don't know about the caribou thing but if your parents are flexible about the time they go, I would think fishing could be done during caribou season.
6. Some types of game in AK (caribou may be one of them) require the use of a registered guide when hunting them which is pricey.
7. Be VERY SURE you don't by mistake put any ammo, knife, etc. in you carryon luggage or you will spend quite a bit of time answering questions by TSA.