As some others have noted, centire scopes work well on rim fire rifles with a few caveats and comments:
1) Parallax is normally adjusted to 100 yards on a fixed parallax centerfire scope, versus 50 yards for a rimfire. However, if you're planning on longer range shooting with a rimfire, 100 yards is ideal.
2) Parallax error results from your pupil being outside the center of the scope's optical system, however parallax error also tends to be minimal at low powers. That's why you don't see many adjustable objective scopes until the maximum magnification exceeds 9x.
3) Provided the .22 rifle in question is adult sized and fits you well with a scope mounted on it, your eye will be both well centered in the scope, and consistently placed. Both will reduce any parallax error.
4) The major benefit of a rim fire scope is the ability to focus at short ranges (10-25 yards or so). A "high" magnification rimfire scope like a 3-6 variable will have an adjustable objective that also allows for the close focus. However with a centerfire variable scope, you can almost always get sharp focus by dialing the magnification down to the lowest setting, even at 10-15 yards.
5) Rimfire scopes usually come with slightly smaller objective lenses and shorter tubes, so you may see a shorter 3-9x33 instead of a longer 3-9x40. But the size difference isn't an issue on an adult proportioned .22 LR rifle.
6) Still, you may find a 2-7x33mm variable scope to be better proportioned than a 3-9x40mm variable scope, and in fact for field use a 2-7x variable scope is almost ideal.
----
I own exactly one rimfire scope (a Leupold 3-9x33 EFR) and after using it side by side with the VX-2 and VX-3 3-9x40mm scopes on my other rimfires, I just oped to go the 3-9x40mm route, with a couple exceptions on smaller rifles where I use a 2-7x33mm scope.
Optics wise the VX-2 was a best bang for the buck series, and the VX-3 series was a really great scope for about $100 more. I have not bought a VX-Freedom scope yet, but it appears to be in the same sweet spot between high quality and excessive price.
I've never been a fan of the VX-1 or Rifleman scopes, especially when you are using the knobs to adjust for elevation and windage. .