First I will let you know I work on Wall Street so take my advice with whatever size grain of salt you would like. I'm a software developer and have never worked at either of those firms.
When an account is transferred from one firm to another the receiving firm will use a process refered to as ACAT which is similar to ACH between banks but for securities, it is slightly more complicated but it allows your assets to be transferred in kind without being sold and the transaction history ca sometime transfer along with the assets depending on the firms involved. The process is either fully electronic or paper based depending on the sophistication of the fit a technologies and their relationships. MS and JPM would be fully electronic and most likely transaction history would transfer.
Now for taxable accounts that is very relevant as it allows you to switch brokerages without incurring taxable events, in your parents case the account is the proceeds of a 403(b) so there would be no taxes but it just eliminates sales fees etc.
Now Finra calls these advisors Registered Reps, they are employees of the firms they represent, your parents are actually clients of the firm not the advisor, it is illegal and unethical for an RR to take their client list to a new firm when they leave. That being said, most people follow their advisors and have no loyalty to their firms, this is a very gray area and subject to lots of lawsuits. Many times the advisors bring the clients to the firms and it is hard to prove they were solicited to leave etc...
Now whether JPM is actually instituting the policy you indicated or not for a certain group of clients or advisors is also completely something you will never know, but you parents might have called them on it and they waived the policy etc...
Now to your actual question, your parents advisor gets paid a fee to advise/sell product, their interests only partially align with the investors. I have no idea why your parents chose to leave fidelity in the first place but they offer free advice in their offices which rangers from lack luster to above average and fee based advisory services that are decent. Every here loves vanguard for the cheap fees, I think there customer service lags behind fidelity and you can get cheap funds with some advice there.
Now Vanguard and Fidelity are great for the DIY investors but some people just prefer an advisor even though it cost them. Depending on the portfolio size really drives what you do. My guess is the portfolio is not 8 figures or JPM would have looked at and made more an effort to get someone more experienced in touch ASAP. So you are in the 6 or 7 figure range, if it's 6 figure go the DIY approach with an advisor at Fid or Van... You will be fine, if you are in the mid to high 7 figure range I would recommend Neuberger Berman (disclosure, I used to work their when it was a Lehman subsidiary, and still have many friend there), all they is asset management for high net worth clients. I used to run test reports for my favorite celebrity clients when deploying new functionality, when you had to pick an account why not a rock star.
-Mister FancyPants