Focus on life, not death. While the topics of estate planning and death are intertwined, it doesn't mean that death has to be the focus of the discussion, rather focus the discussion on planning to live well, instead of planning to die well.
Encourage your parents and other family members to discuss their wishes. It is important that your family discuss what they want to have done and who they want to do it. Ask your family what it is they would like you and/or others in the family to do in the event of an emergency.
The need for protection for the entire family should be emphasized. A formal estate plan helps to protect the entire family from creditors and possible predators.
Encourage parents and all family members to seek expert legal and financial advice. By YOU doing so, you are setting an example while at the same time sending a message that you trust everyone else to handle their own affairs, this will more than likely help them breathe easier by knowing you don't want to control the situation. Offer to help them with their search for a qualified attorney.
Be patient and understanding. Be realistic, the initial conversation with parents and other members of the family about estate planning will not be easy and it will not be, nor should it be, the last one. It may take several discussions before even a little bit of headway is made, and even after estate plans are in place, it does not mean that these plans will always remain relevant in the future, it is an ongoing dialogue and process.