If a player has the following permissions:

• helicontrol.callheliself
• helicontrol.cooldown.5
• helicontrol.limit.3

using /callheli says they don't have permissions.

/callheli works just fine having helicontrol.admin.

I tested by removing helicontrol.admin from myself and assigning the permissions mentioned above. The /callheli command said I didn't have permissions.

Presumably you mean you're running /callheli with no arguments, and if so, this is intended behaivor. This command calls in a helicopter. Not to a person, not at a specific spot, but just as if it were any regular timed helicopter spawn, so it'll fly into the map, and this is why the permission is separate. If you supplied your own name as an argument for /callheli, it should work.
If you want to be able to call it in the manner where it just flies into the map, grant the "helicontrol.callheli" permission. Should work, and shouldn't override anything else. :)

Ok, thanks for the fast response. My bad.