Cream butter with electric mixer. Add sugar and cream until fluffy.
Add eggs and cream until mixed.
In separate bowl, combine flour and baking powder.
Add flour mixture, grated lemon rinds, vanilla and rum to butter/sugar/egg mixture and mix until combined.
Cover and refrigerate over night.
When ready to bake, preheat pizzelle maker.
Add small amounts of batter to pizzelle maker. Quantity will depend on how large your pizzelle maker is. Start with 1/4 cup of batter. If the batter oozes from sides of pizzelle maker, you've added too much. If it doesn't completely fill pizzelle maker, you've added too little.
Bake cookies until golden. Remove to plate or wire rack to cook. Repeat with rest of dough.
Like all pizzelle type cookies, these take a lot of time to bake in those makers. But they are worth the time!