Your dad is wrong.
And the first sentence does not have to be read as a "conditional," but as a statement of fact, with if meaning since.
Since my dog doesn't like you, (then) neither do I.
Even if we read the first sentence as a conditional, it is talking about a present real situation, so the present tense is used in the "then" clause. The driver doesn't know if his dog likes you or not but if the dog doesn't, then neither does the driver. This is not talking about a hypothetical situation, but about a real situation. The driver, in this conditional, is not sure whether the dog likes you or not, but whether the dog likes you or not is an actual (not hypothetical) situation.
I wouldn't use would in the "then clause" unless the sentence was talking about a hypothetical situation:
If my dog didn't like you, then neither would I.
Here, the if-clause is not taking about a real situation but an unreal one, and thus the sentence is hypothetical. In actuality, the dog does like you. I am talking about a hypothetical situation in which my dog doesn't like you.
Another interpretation of this conditional is that it is talking about a past real situation, and what I said about the present real condition applies here, except in the past. The use of would in this case refers to repeated or habitual action in the past.