Amended tax return to show I am a dependent, but parent A's tax came back rejected. Do they have to refile their return?

If their return rejected because you claimed yourself on your original return then they will have to print and mail their return now.  Even if you amend it won't clear your ssn from the IRS.  Once a ssn has been used you can't efile it again on another return.  

It might take months or a year or longer for the IRS to use the amended return to change the records and allow your parents to e-file.  The IRS has millions of amends and will get to them eventually.  If your parents print and mail with a note explaining what happened, it might be quicker.  But either way, I would guess this will cause a long delay in processing their return.