Air Parity, as you note, mean no D-Day. Even air superiority was a question. The WAllies want air supremacy. They wanted to be able to truthfully tell their troops that if the looked up and saw planes they belonged to the Allies.If the LW has anything close to air parity something has been going wrong for years.
To Quote Wiki
To have enough aircraft to match the Allies over France - and given the demands of the Eastern front they would needs 10,000's of extra Aircraft.
How are they going to find the fuel for all those?
If the Allies had not been winng the Airwar in late '43 early '44 they would have sought to ramp up aircraft production and crew training until they were. Germany can not hope to match Allied production capability.
Luftwaffe needs, at minimum, 1,500 fighter to cover at least that many fighter bombers and medium bombers capable of responding to the inital landings, and at least 300 night fighters to winnow the airborne stream over the Channel with triple that number to replace losses; this assumes that the WAllies don't simply pause the U.S. portion of the CBO for a few weeks so they can throw an extra 1,800 fighters into the air battle and devote the bomber groups to airfield interdiction. In reality Luftflotte 3 (Luftwaffe forces, France) had a TOTAL of 481 aircraft, 100 of them fighters and 63 recon aircraft, while the WAllies total muster of fighters alone (include 8th AF Fighter Command) exceeded 5,400,
If by some miracle the required number of aircraft appear (and it would have to be mainly from Inner Germany given how much of the Luftwaffe's single engine and night fighter strength was concentrated there to oppose the CBO (TOTAL Luftwaffe strength on the Easter Front in mid 1944 was 408 single seat fighters, 97 Night Fighters, and 610 ground attack aircraft, or barely enough to fight it out over France the first couple days if sent West). the WAllies would have postponed until the new forces had been destroyed using the same sort of Hammer & anvil tactics that had already done so much damage to Luftwaffe strength.
As long as the WAllies have air supremacy and naval supremacy in the Channel, especially having enough ships for the gun line, the invasion can NOT fail once the landing force meet up with the airborne a few miles inland. With that much depth and overwhelming airpower and big guns afloat (and pretty much every gun on a warship is big gun in ground component terms (a single destroyer reprents as much firepower as a battalion of 105mm guns and a light cruiser is a regiment of 155's, while battleship fire is like the wrath of an angry God).