Parallels requires a true-blue copy of Windows which it runs in a virtual machine on the Mac.
CrossOver does NOT require a copy of Windows but seeks to emulate the Windows environment that programs expect.
They are thus very different animals.
Parallels will pretty much run any Windows program (as expected)
CrossOver (and Wine) is getting better and better, and a lot of Windows programs
will run under it, some more successfully than others.
But MS keeps moving their proprietary goalposts so CrossOver/Wine will always be a few steps behind in terms of compatibility.
Mac