Asks for computer password before putting programs in garbage can

If I may...
Please delete Adobe Flash from your computer.
Chrome has a Flash emulator, no need to run Adobe Flash on any device at this point, besides introducing new pathways for malware and viruses.

Now to fix you.
Password to delete is required in a couple of instances:

1. You have password protected a folder
2. You are not the owner of the file and/or admin of your computer

Verify these two items first.
Make sure the file folder is not password protected.
Make sure you are the owner of the file.

I see that you have Bootcamped your Mac. Ensure Windows10 is using the default virus protection and not something like McAfees or some other third party virus protection. Windows10 has come a long way in keeping your computer safe.

 

I removed Flash from my 2009 27" iMAC some years ago and never missed it. The Chrome emulator worked just fine in the few instances where I needed it. When I bought my 2015 iMAC Flash did not come pre installed and I never downloaded it. The "Flash Update" you see is a very common virus attempt and the notification that you have a virus is the same. I get the Flash Update periodically and the virus warning comes in various forms as well. It is a better world without Flash.

 

OP:
Download MalwareBytes for Mac and run it:
https://www.malwarebytes.com/mac/

It's free.
The app will ask if you wish to upgrade to the paid version.
YOU DO NOT HAVE TO DO THIS.
Just run it in "free mode".
After 30 days, it will "convert itself to free, forever" and keep running that way.

An excellent piece of software.
Run it, and report back here with the results you get.

 

"A couple of days ago I was in Best Buy at their Geek Squad and they will wipe the MBA and reinstall macOS High Sierra for $100. While I would rather just get rid of the Parallels partition on the MBA, I'm tempted to go the Best Buy route. I realize I would have to back up my data before a complete wipe."

Why pay $100 for something you can do yourself?
Really, why...?

 

While I'm all for paying a professional for their time, wiping a MacOS is certainly in the consumer realm. Being charged $100 for pressing three keys and a bit of internet time is a bit cheeky to me.

https://support.apple.com/kb/ph25649?locale=en_US

As you said, be sure to copy important files to iCloud or some other backup solution.

 

MacOS has a protected recovery partition (section of the hard drive) that stays intact and doesn't get wiped when you reformat. That's how you're able to reinstall the OS without having to use discs or plug in an external hard drive. It's all actually way easier than it used to be with swapping discs and all that. Once you start the process, you can go have that Ruth Chris steak you just paid for by not going to Best Buy. The install will be done before dessert.

It will definitely let you reconnect to your wifi during the reinstall process, no problem.

(And this should go without saying, but you obviously want to save any important files to some other drive before you do this because at the end your hard drive is going to look like it did when you first got the computer new.)

 

As stated above, never do a flash update suggested by some obscure website from a link. If I receive that message I go to system preferences and do the update from there:




Quick search seems to indicate that npapi is the older plugin.