I have two boys; 7 & 4 years old. I also have 5 dogs (2 big, 3 little) & 2 cats. I have my tiel & my ysgcc. I would say it's not as much if the bird can live with your family, as it is if your family is ready to live with a bird. Conures are very social, loving birds. They need lots of attention & an hour minimum out of cage time each day.
Your children will obviously be excited about the new family addition, but need to understand giving a new bird their space is imperative.
You know your children & if they are ready for the patience a new bird requires.
To answer your question simply...a bird is very family oriented. Your children & dog will most likely add to the enjoyment your cinnamon would have in your family. It could take time. I've had my birds for a few months & everyone is still learning & adjusting; humans, furr-babies, & feather-babies.
Good luck with your decision. It's a very exciting one to make!!