I don't want to be rude to your parents, but I don't think you should put your life and ambitions on hold for dogs. As much as you and your parents might have a great relationship, you are a grown woman and you can't be held back just because they need a dog sitter. They have many months where they could work out other arrangements, maybe even having a couple different people take turns keeping the dogs.
You aren't being selfish for wanting to go to school, it's something that will further your career and improve your quality of life. You are a grown woman, and sooner or later you will need to move out, and become independent of your family. The longer you wait to do it, the harder it is to do. If you wait another year (or more) to start on this, that's even more time until you reach your goals, and another year of wasting away at a job you don't like.
You need to do this for you, to take control of your life, and do the things in life that you want to do. Don't sit back and wait to see where life takes you, be proactive and make it take you where you want to go.