Great question! The answer is a resounding yes! Your dad needs landlord insurance because he owns the home but you live there full-time. If your dad were to move into the home full-time, he would just need home insurance and he could drop the landlord insurance.
Landlord insurance covers the home and personal liability, but it does not cover your personal belongings as a tenant, such as your clothes or computer. Even though you are a family member and you do not pay rent, you are still considered a tenant for insurance purposes. Therefore, you still need renters insurance to cover your personal property.
The good news is, your dad does not have to pay for both landlord insurance and home insurance at the same time; it just depends on whether he lives in the home full-time or not.