Since Family (to me at least) is such a typical response to a question like this. I would really make sure the response stands out among all the other people who chose to respond with family.
The parts that I think that makes your response strong is how your mom took classes so she could help you, the last bit about your dad, and your final sentence.
The part that I don't understand in your response is this:
When the summer we decided [...] hardship they did growing up.
I don't see the connection. How does your family moving to Vancouver related to sacrifice of your parents living away from each other? Did one of them stay in New Brunswick? That area needs to be more clear because I got confused.
Also I would identify where "here" is for My family immigrated here in 2009 because you said you originally lived in New Brunswick then moved to Vancouver. Your "here" seems to indicate Vancouver which then doesn't make sense.
"and seeing snow for the first time" sounds like an incomplete sentence.
I think it should and saw snow for the first time
Aaron, using your family as a reference to what is important to you is incomplete. It is not your family that is important to you but rather "time spent with my family". Those are two different topics for discussion. In this instance, you should clarify that you value "time with family" and then explain why. Discuss how and why it has been impossible for the 5 of you to get together for any reason. Then explain how the times that you do spend as a family is extra special because of this. What do you do as a family? How have these limited bonding moments helped to strengthen the relationship of your family? After explaining these things, you can then pick out what to you is the most important factor when spending time with your family and why. Your general statements at the start need to gain a more focused presentation towards the end in order to create a strong and informative statement which the reviewer just might remember.