score:2
There is not a lot online on how pets are considered for ties to home country, but the fact that they are never mentioned in the lists with options shows they are not considered any strong tie.
While you feel the dog as a family member, immigration officers do not. Likely out of experience as too many people would not return home to get back to the dog and because it is possible for the dog to be send to you while you are in the UK. It is not as if it needs a passport and a visa.
You should mention your parents if they are an influence in your position, but if they do not depend on you nor you on them they will not be considered a strong part in consideration for your case.