What definition are you using?
USAAF definition is :
To be superior in the air, to have air superiority, means having sufficient control of the air to make air attacks on the enemy without serious opposition and be free from the danger of
serious enemy air incursions..
Here are some more definitions of these terms recognized by Military professionals:
"Air supremacy is defined as the condition when the enemy air force is in-capable of effective interference. Through the complete destruction of the enemy air forces, this condition is the ultimate goal of an air campaign. Yet, this condition may be difficult or even impossible to achieve. It may occur however, through the establishment of a diplomatic “no-fly zone”. Under the condition of air supremacy, the air commander employs all of his aircraft at will."
IMO this condition was achieved by the Allies on a few occasions for a short period of time in a local geographic region. IE Normandy.
"Air superiority is defined as the condition when the conduct of operations is possible at a given time and place without prohibitive interference by the enemy. This is a necessary and obligatory condition to attain success in combat and overall victory in a war. The most efficient method of attaining air superiority is to attack enemy aviation assets close to their source at maintenance and launch facilities, early warning and C2 sites, and ground-based air defense sites."
IMO the Allies got and maintained Air Superiority from Mar '44 til the end of the war.
"Local air superiority, which is purely geographic in nature, is characterized by well timed air attacks to coincide with enemy aircraft downtime, re-turning sorties, aircraft rearming, or gaps in air defense coverage. This condition may also occur in sectors across the theater of military operations where the the enemy may not have adequate assets available to ensure air superiority. In certain situations or against certain enemies, local air superiority for a specified period of time may be a more realistic goal."
IMO This is what the LW was able to do until late in the war over the Eastern Front and in such Operations as Bodenplatte in the West. The LW failed to recognize the threat of the Western Allies Air Power until it was too late. This threat was NOT lost on the pilots of the LW. The Leadership at the highest levels failed to recognize it.
"Air parity is defined as the functional equivalency between enemy and friendly air forces in strength and capability to attack and destroy targets. Under the condition of air parity, where neither side has gained superiority, some enemy capabilities affect friendly ground forces at times and places on the battlefield. Air parity manifests itself to the commander primarily in the amount of fixed-wing aircraft used for direct support of ground forces. More aircraft are dedicated to interdiction and strike missions to gain air superiority. "
IMO This condition did exist for short time periods after Normandy but was a tactical occurance over a specific point on the map and not a strategic one. IMO Strategic Air Parity existed from late '41 til Mar '44. During this time Niether side was able to launch an effective offensive and both sides pretty much equally attempted various offensive measures.
Here is another good article reference Allied Air Superiority over Europe. The Author uses the term Supremacy but I think it should read superiority.
http://www.butler98.freeserve.co.uk/thtrlosses.htmCrumpp