"I’m kind of new to scopes, so is there something more to correct parallax set up other than making sure the target is in focus as well as the reticle? Also, how does one know for sure if the scope is mounted correctly, ie. not torqued? Your input is appreciated."
The most important part will be to check that lower part of bases are torqued tight on your rifle, then level the rifle and put the scope and align the vertical line of reticle to a cord with a weight so it is a straight line, use the corner of an open wall, etc. If it were me doing the mounting, I would use an aligning bar I machined for making sure the bottom bases are perfectly straight and not torqueing the tube of the scope, and it they were, I would lap them...But let's assume that those bottom rings are aligned...
You have your scope over the bottoms part of the rings aligned...Then put the upper part of the rings...Make sure you use those uppers as they came from the store you bought them, depending on the quality and brand, some of those are machined and while you can mix them don't do it because they are matched... Put some masking tape and mark them.
Put the upper parts of those rings and torque those screws making sure the gap in between the bottom and upper parts are about the same...Torque those screws to 40 in./lb and if you don't have a torquimeter, then do it to where you feel good resistance and then 1/2 turn more...Use blue lock tite (never use red) so no screws come loose.
For checking your parallax, align the HORIZONTAL of your scope to a horizontal line at the distance you are shooting and then adjust your parallax knob on your scope, the horizontal of your scope and the line on the target should loos as 1 line...Then move your head up and down and see if that line moves, if it does you need to adjust that parallax knob until it doesn't.
Clean your gun, fill it to the max pressure and shoot...With a clean and full of air gun expect a few 3-6 shots maybe going to different places...This is normal while the regulator (if regulated) or the valve (if not regulated) set and also while the barrel gets a little lubricant from the pellets...After 6-8 shots shoot several 5 shot groups...
If your gun keeps on moving the POI, then you have a crown that may have a burr or is damaged, or your barrel is touching something (if fully floated) and if not floated, the touching point may be too tight or too loose...Check this as I have encountered innumerable problems caused by this issue (over and under pressure in the touching point)...The last thing to check will be your stock and action...They should be tight..Adjust them to the same torque as the scope.
If your problem persists then check your velocities with a chronograph, if variations are within 1-2% of the max velocity you are shooting, then you are fine...The last step are your pellets: Try heavier and lighter and try different brands...Try: JSB's and H&N Barracudas in different weights...You can also try Crossman Premier (The ones that come is the carton color box, I believe 1250 pellets/box).
If still have a problem, then you really have a problem and that could be your barrel is not good, damaged crown, or ????...Someone needs to shoot it and eliminate all possible variables.
Note: I am assuming that you have a very steady table and rest and that you have enough shooting skills to be consistent.
Please let us know your findings and how you are doing....
Regards,
AZ