Yes, an ADR at Tusker House (or any theme park restaurant) would require the use of a park ticket. If you have the Park Hopper option on your tickets, then you can visit another park during the morning and evening and just "hop" over to AK for lunch.
You can NOT use two days of tickets (no park hopper) on the same day. In other words, if you go to MK in the morning, you can't use another "day" on your multi-day ticket to get into AK in the afternoon. You would need the park hopper option.
Resort dining wouldn't have this restriction nor would dining at Downtown Disney.
Good luck with the planning!