I used to work in Operations for a mutual fund custodian (basically operations for about 20-30 MF companies). Most of your questions have been answered, PM me if you have any other questions.
To sum up
They have an RSP account.
With money invested in funds. Lets called it the vested part of the account.
If they were to transfer money in the account, without investing it, it would be referred to as the cash part of the account (which you understand already since you trade.)
If you sell the vested part of the account, it'll go to the cash portion of the account. At that point you can whether
a) re-invest it in the funds/stocks you deem best for them
b) submit a registered transfer form, sometimes they're simply online, to transfer to another financial institution (which from what i understand your parents don't want to do) and all this process is tax free. SOMETIMES they might have transfer fees, sometimes the receiving institution covers those fees, depends on the policies and promos at the time of the transfer.
In both cases, if you haven't done so, you might want to get Trading Authorization on their account first though as usually the reps might not be so helpful if you have questions about the account. Even though i'm guessing you just get them to log in and do the work for them

And you probably know all of that but ya know... just making things obvious lol
Anyhow, in any scenario, definitely check the MERs of any funds you get, some of them might have higher MERs but have the returns to somewhat back it up some have lower MERs as they're only offered through certain platforms (such as self-directed investment accounts where an actual broker doesn't need his cut

) Definitely check if those funds have Deferred Sales Charges (back-end) before selling or if they're not applicable (like front-end funds/no-fee funds)
I think that's all i can think about for now