Vinay3-
You are right that parallel clauses should start with the same word. The issue is that the parallel clauses in your example don't start with the word "that". Instead, they start with the world "would." Think of "that" as the last word in the first part of the sentence, before the parallelism begins:
I told my sister that I liked her dog and I hated her cat.
I told my sister that:
- I liked her dog and
- I hated her cat.
*We don't need to repeat the word "that" here because it distributes.
Given the chance, I would move to Hawaii and open a surf shop.
Given the chance, I would:
-move to Hawaii and
-open a surf shop
*Again, the word "would" here distributes to both verbs. Notice in this example that one clause starts with "move" and one starts with "open" because they both actually use the word "would."
The idea behind our rule that parallel clauses must start with the same word is that you can't have a situation such as the following:
-I bought a house that has four bedrooms and where I can relax.
Here, one clauses starts with "that" and one starts with "where." This is not OK, since these words are serving the same purpose and to make it parallel you'd have to be distributing the word "house," which you can't do.
Make sense?
Brett