I am at wit's end trying to install Windows 8.1 on my 2009 iMac using Parallels Desktop 9, and the new virtual machine will not accept my product key and suggests I check my installation media. Are you familiar with this issue?
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That can be a problem.  Yes, I’ve encountered it before.  I see you’ve made a phone appointment for tomorrow.  I suggest you first get on the phone with Microsoft.  There is a 1-800 number you should be able to find on the Windows 8.1 packaging.  If you’ve used the install before on another machine that can be an issue.  There is however, a way to (over the phone) be issued another install key.  You can answer a couple questions, one of which is something like “are you reinstalling Windows after a hardware failure”.  If you say yes to that, you are usually issued the new product key.  You’ll have to jump through their hoop of writing down an incredibly long series of numbers and letters before they issue the new key.  It’s a total pain but then again, that’s why Microsoft is in the position they’re in at the moment.  By the way, Windows 8 is very difficult to use on a non-touch device.  it’s basically a failure.  It works fine on touch devices but is difficult to use without a touch screen.  If you can stick with Windows 7, I would encourage that.  Just my opinion.

The above scenario is only one possibility.  It’s also possible your version of Parallels doesn’t yet support Windows 8.  If you’re moving to Windows 8, you need to be sure you have the latest version of Parallels and all the updates have been run.
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Update:
I have been on the ph with Microsoft 3x now in the last 7 days and no one will just give me a new activation code, much less speak English I can actually understand. I get passed from person to person and the issue does not get resolved. I have Parallels Desktop 9 so I think it will run Windows 8.1. My ultimate goal here is to be able to run a version of Quicken that isn't made for Mac (Quicken Home & Business) that meets my needs. Happy to use Windows 7 which I suppose means I have to get a refund for Windows 8.1. That's my only purpose for having Windows at all. 
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Overall, I think you would be happier in Windows 7 however I don’t believe it will be easy to return open box software (windows 8).  When you try to install you should be presented with a phone number to activate over the phone.  Microsoft has many phone numbers depending on the software you’re trying to activate.  Here are a few articles Microsoft provides that may or may not be of use:

Why can't I activate Windows? - Microsoft Windows Help

Activate Windows on this computer - Microsoft Windows Help

Unfortunately, being Microsoft, I can’t guarantee that I’ll be able to solve your issue in this case.  But I am willing to try.  Microsoft is just so hard to deal with.  Let me know if you’d like to keep our scheduled session later today.

Update:

attached are screenshots of what goes on. It is an iOS file as you can see in the attached within Parallel Wizards window. Link to my login provided.

Just the link is not going to work.  I would actually need your login email and pw.  But the picts you sent look like you are doing everything correctly.  Can you tell me how large the ISO file was?  select it and hit command+i and it will tell you file size.  

I doubt this is a Parallels issues.  This looks to be a Microsoft issue where the key is simply invalid.  There is a possibility that the ISO is an update only and not the complete OS.  The file size will give me a better idea of that.  Another option would be to burn the ISO to a disc then install that way.  Although it’s probably not going to make any difference.  I’m going to do a little more research on the price you paid.  $119 seems low to me.  

When you hit next after the product key fails, does it give you the option to authenticate over the phone?

I feel your pain.  I’m sorry you’re having to deal with this.

Price looks right for the basic version of Windows 8.  This seems to be a simple case of an invalid product key.  Microsoft is going to have to issue you another key.  Simple as that.  I bet that will solve the issue.  The problem is getting them to issue you one.  If you use the “activate over the phone” option during the install and the product key you have is rejected over the phone, then you’ll have to contact support via email I’m guessing and describe your case that way.
You could also try downloading the ISO again however just in case the original download was corrupt but I doubt that’s the case.  I’m just reaching now.  

There’s also the possibility that Parallels is expecting a Windows 8 Pro product key but you have a standard product key.  Did you make sure you are selecting the standard Windows 8.1 install and not the Pro option.  Also, make sure that all Parallels updates have been run.  Use “check for updates” within Parallels.  Windows 8.1 was just released and that could be an issue.  Parallels may be expecting 8 but you have 8.1.  Just sharing possibilities.