The cap certainly has an impact on parity... though it is less about the average team and more about the extremes. There are ways around it, but it means the gap between top and bottom teams is controlled. The Rangers pre Cap were spending 75+m, Toronto 65m, Avs were 60-63m, Dallas 70m... while some teams were around 20-25m. We're talking about teams spending 3x what the bottom teams spend. So if Arizona was spending 50m today, the Rags may be at 150m. Narrowing that difference certainly creates some parity. Frankly we are feeling that parity today... if the Avs had unlimited funds to spend, Kadri would be here, so would Burkie, Compher would be the 3C, etc etc. The fact that the Avs (and pretty much every good team) has to lose players creates parity.
That's not really why the cap exists though... it is exists for cost certainty. There isn't a realistic way to consistently being competitive with a payroll that is 1/3 the top teams. If you're not competitive, attendance normally suffers to a large degree. Which makes markets quickly fall out of favor. With a cap, teams should be able to mostly create a competitive roster... at least to the point where they get into the playoffs every so often for the extra revenue. They can also budget and plan for costs over a longer term. Instead of having a random huge year. Like the Avs and Sakic... which caused an ownership change... and funny enough without Air Force One, history is far different. Under the cap surprises can happen, but not to that extent and they are more manageable.
The cap has worked quite well for the NHL. Outside a few markets, business is very good. That is not even close to what the case was in 2002 where half the league was on the verge of collapsing. I get that it causes some issues with good teams and having to break up players... but without it, there is likely a bunch of ownership changes, relocation, and likely a much smaller revenue.