How can I find the purchase date of my iMac?

The purchase date is not the date you need. What you need is the year designation of your Mac that's in the About This Mac window:

That's the date designation that will determine what you can upgrade the system to.

I would try

https://checkcoverage.apple.com/

I do not know my purchase date for my iMac 21.5 mid 2011 computer so I cannot access any upgrade downloads