I'm not going to allow my partner and son to have a dog
Why won't you let them have a dog?
I can't deal with dog hair.
Getting a dog is a big decision and everyone in the family should be on board with it. If you don't want a dog, then it's up to your partner to respect your wishes. However, this may be something that you want to compromise on because if your partner wants a dog and you don't then they may resent you.
Why should I care if they resent me? I am the parent in the house and I want dog hair out of my life. My son is 10 and I'm not opposed to him to grow up around dogs. If they want to get a dog I'm going to insist they get a small dog that sheds and poops less.
That's wrong to not care your behavior is causing resentment. But I'm glad to hear you're open to compromise. You need to discuss this with them from with a conciliatory frame of mind and drop the "I don't care" attitude.