Because the subject does not have to preceed the verb as it does in English. If you wanted to emphasise that it is pets that children don't like you would put pets first in the sentence. There is nothing in the sentence to indicate which is the subject and which the object of the verb.
So, it's an interesting question: if a german speaker wanted to make sure that Die Tiere was the subject and not Kinder how would that be accomplished? I can't think of an obvious way. Maybe something like Die Tiere haben Abneigung gegen Kinder - but that sounds obtuse to me.
You just go by context and tone of voice. The SVO order is generally more common than OVS, especially out of context. Sometimes other parts of the sentence, like adverbs, will indicate which it is... Also, it's easier to tell when the verb is conjugated differently for each noun for obvious reasons. Like if it were "Haustiere mag mein Kind nicht" or something like that.
I grew up and still live in Central PA. There are still people here that speak English with some German grammar but obviously without the der, die das. It's not that odd if you're used to it. But if you are looking for the subject to always be at the beginning of the sentence you my be confused. We just speak in a manner that is more inflected especially if we are asking a question.