You are both being unreasonable. However, I suspect this is only partly to do with the presence, or otherwise, of your sisters dog.
OP, I can understand why you would rather the dog be kept away given recent events, but I think it a bit OTT to cancel the visit altogether over this issue. Surely a better compromise would be to ask for notice of when DS and dog are coming round, and go out? Not ideal, but as you do recognise your fear is irrational, perhaps better than not visiting at all?
I think it is also OTT to accuse your parents of prioritising a dog over their DGS. They aren't- if anything they could be said to be prioritising the your sisters wishes over yours. That is, I suspect, what is upsetting you- that your parents are not "on your side" on this and are not allowing your fears to over-rule your sister's normal routine/wishes.
Is it possible your parents are fed up of having to "take sides" and just want their DDs to sort it out themselves? Let's face it, whichever "side" they come down on, they are going to upset someone.
Your sister is being unreasonable in the sense that she could leave her dog at home, and I doubt it would make much odds to the dog or her in the grand scheme of things.
Perhaps she doesn't understand how upset you are. Have you told her? Maybe she thinks you're being a bit of a drama queen, and doesn't see why she should bend over backwards to accommodate your demands, but if she knew how upset you were she would back down? Or perhaps she just doesn't see why she should disrupt her normal routine over what she sees as something totally irrational- perhaps not the most considerate thing to do- but I can see why she would hold that view.
I think you sound quite dismissive of her sister- and her parenting. Maybe you have good reason, but I wonder if your sister senses this and doesn't see why she should alter anything on your say so- especially something she sees as irrational- so is digging her heals in.
You say your parents won't stand up to her, as if she gets annoyed she won't let them see their DGC. But aren't you going to be doing the same thing? You are saying you won't visit if they don't do what you want- so if your parents don't do as you ask, then they won't see your son. I do note that they have not been to visit you in your home either though.
My advice, OP, would be to go and visit your parents and arrange to go out when your DS comes round. I think you also need to work on getting over your fears, as it's not worth letting it get out of hand. Oh, and suggest your DPs visit you in your home next time.