Yes, the order is important. The parents' return must be amended first to remove the student as a dependent. Then the student's return can be amended.
If you wish, you could prepare both amended returns and send them by mail to the IRS in the same package. That way, they should both be processed about the same time. Otherwise, you should allow 16 weeks or longer for the IRS to process the first amended return before submitting the second one.
It is difficult to say whether this is a drastic amendment to make. As long as you are within the boundaries of the tax law and the time limit to file an amended return, you should be fine.
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