Your question has two parts.
Firstly, to address the issue from a purely equipment point of view.
The short answer is purely on "qualitative parity" the Russians had this in 1941 but once the Panther and then Tiger entered service, they lost it and never regained it. I will elaborate.
I will mainly concentrate on medium tanks as that is largely what decided the war from an armoured point of view.
At the start of the war, the German tanks weren't particularly strong, but the way they were organised was key to early victories. By having large numbers of tanks concentrated together, with support from the Ju 87 Stuka in particular, they were able to use Blitzkrieg tactics to great effect.
However, where they met enemy armour in any numbers they did struggle, the Battle for Arras is a good example.
The Germans had to use Flak guns to counter the Matilda Mark II tanks of the British forces as the Panzer I and II could not penetrate their armour. The French Char I was also more than a match for the Germans in a one-on-one fight but the Germans usually had such a numerical advantage that they were able to quickly defeat France in 1940.
In the early stages of Barbarossa, the Russians had access to a few T-34s that were superior to anything Germany had. The Panzer IV (best available German tank at the time) was a fairly good tank but older in design than the T-34 which had sloped armour, better mobility, more powerful gun etc.
At the start of Barbarossa there were around 1,000 T-34s in service although a lot of these had fairly inexperienced crews as they were a new type and the 3,300 or so German tanks were able to deal with them although they did have some trouble. This is a quote from the 2nd day of Barbarossa from a German battle report:
Half a dozen anti-tank guns fire shells at him [a T-34], which sound like a drumroll. But he drives staunchly through our line like an impregnable prehistoric monster... It is remarkable that lieutenant Steup's tank made hits on a T-34, once at about 20 meters and four times at 50 meters, with Panzergranate 40 (caliber 5 cm),[nb 1] without any noticeable effect.
— German battle report, Finkel 
As the T-34 became more numerous and the crews more battle hardened, Germany realised they would need a new type to counter this opponent.
The Panther entered service in 1943 and was intended to counter the T-34. In many ways the Panther was superior to the T-34 but it was less reliable (largely due to being very complex) and also cost a lot more to build - figures vary but it is generally accepted that the Panther cost about 3 times as much as the T-34. For this reason only 6,000 Panthers entered service during the 1943-1945 period compared to 16,000 T-34s in 1943 alone!
On a one on one fight, the Panther was more than a match for the T-34 but the Russians usually outnumbered the German tanks by some margin.
Panther had better armour, up to 120mm compared to 60mm for the T-34, more effective main gun (both had around 75mm but the Germans had better systems), speed was similar but the Panther had better suspension so that made targeting easier. The Panther was also bigger at 44 tonnes compared to just 26 for the T-34 so it's almost a heavy tank versus a medium tank, no competition really especially when you consider the Panther was newer and designed primarily to be a T-34 beater.
The Germans also developed the Tiger (I) tank around the same time to counter the T-34 and the Russian heavy tank the KV I.
Only 1,300 or so were built in the entire war and so its effect was dubious as the quantities were simply nowhere near enough to counter the Russians with their tens of thousands of T-34s and 5,000 or so KV1s.
The Germans repeated the mistakes they made with the Panther and ended up with a very expensive, hard to maintain and comparatively unreliable tank.
The Tiger was more powerful even than the Panther but at 56 tonnes you are entering the realms of heavy tanks and the ground war was decided primarily by medium tanks.
The Battle of Kursk is generally regarded as key to the Eastern Front and the turning point where Russia took the initiative:
Germany only had 200 or so Panther tanks available from a total of 3,000 tanks to the Russians 5,000 total, largely T-34s. The result was a loss of 1,500 tanks to the Germans
and 1,800 for Russia. So, despite losing the battle the German tanks were able to inflict more casualties than they lost, perhaps proving they were stronger on a per tank basis, but superior enemy numbers was decisive.
On to the second part of my answer, organisation.
The Germans organised their tanks into large groups from the start of the war and used Blitzkrieg tactics, using airpower to support the tanks. This allowed for quick victories early in the war as the defenders tended to deploy their armour in more "piecemeal" fashion.
Russia didn't do this however, but the Russian army was in a bad state at the onset of Barbarossa for a variety of reasons including Stalin's purges that had killed off a lot of their best commanders, lack of training, poor supply etc.
It took Russia a while to reorganise themselves on the German model and once they had done this, couple with their plentiful armour, they were able to drive the Germans back, regardless of their inferior (in a one-on-one fight) armour.
To summarise, Russia won the armoured battle due to having more numerous, more reliable equipment, modelled into units along the lines of how the German's had deployed their own forces, ie in large groups with armour of prime importance.
Had Germany built something along the lines of a slimmed down Panther that was simpler, cheaper and more reliable, they may have been able to produce them in sufficient numbers to change the outcome. But that's a slim chance.
Finally, interesting point of view regards the T-34:
Another interesting article:
I haven't referenced everything I've posted as it would be a mess of links to Wikipedia, but feel free to search yourself!