I have moved many times with different cats and the most important thing to do is to keep them inside for the first two weeks (even though they may scream to be let outside!). After this period of time they should know where "home" is. It will also be of enourmous help if you can reserve a part of the house for your cats so that they feel safe somewhere without having to share it with your parents' pets. As for getting used to the other cats, this can take a lot longer. I have had some cats take up to 3 months to get used to the others. Apart from mediating their initial encounters there is not much you can do but encourage them to get along and stop them fighting.

I hope it goes smoothly for you. Troy.