I think there was a communication problem in speaking with Dish about what you want, etc.
It sounds to me like Dish didn't want to have to "Move" you since your parents already have a working system. In other words, Dish isn't going to pay to put up a 2nd Dish/system as part of Dish Mover when the address already has a working Dish Network system. That's most likely what they meant when they told you they don't "allow" a 2nd Dish; then meant they don't want to pay for it.
If you were thinking of cancelling your account (then you will have to pay the $180 cancellation fee) and just adding your 322 to your parent's account: Dish will only do that if you OWN the 322. They will not add one of their leased boxes to your parent's account. Perhaps that is why they said they would cut service to both accounts, as per your original post. If you want to add a box that you will use on your parent's account, you need to buy your own box at retail; that will be your OWNED box and Dish will gladly add that to your parent's account. Oh, and that situation does not count as a FREE Dish Mover, so yeah they will want to charge you for connecting your to your parent's system, which you might have referred to as the "Moving Fee".
If you want to keep your own, current, separate account (and, presumably, not have to pay the $180 cancellation fee), then the question you must ask is the following: "What must be done so that I can receive the same services I receive today when I move to my parents who have also have Dish Network, and is that covered in the Dish Mover?"
I would hope that since you are keeping your account, Dish is willing to connect your box to your parent's system. This may require some cost to Dish in the form of changed-out LNBF's, external switch, additional dish, or even a new system IF you need access to sats your parent's don't currently see. Then again, maybe Dish is afraid to get you connected at their expense only to have you drop your account, then add an owed box to your parent's account. However, they may want to charge you for that.
Final details: Maximum of 6 (SIX) receivers per residential account. If all 6 are dual tuners, that give you 12 tuner to work with. Only up to 4 (FOUR) of those tuners can be leased, for example only 2 (two) dual tuner boxes can be leased resulting in 4 (four) tuners.
It would be great to hear of some experiences or get clarifications regarding the FREE Dish Mover program when one is MOVING to an address that already has an open Dish Network account and functioning Dish Network system even if one wishes to keep their current separate account and NOT merge account.
Let us know how this turns out. I will be interesting to see how Dish does this.