The Soviets made a GIANT contribution to the success of the invasion, although for their own military reasons. On June 22, they launched Operation Bagration, the largest Allied offensive of World War Two, with 2.3 million soldiers. The offensive crushed the German Army Group Centre, including Fourth Army, Third Panzer Army and Ninth Army. The result was to sharply limit the number of troops and tanks the Nazis could allocate to Normandy.
It would have been nice (for the Western Allies) if Operation Bagration had been launched the same day as D-Day but even if the Soviets had wanted to do so, they couldn't have--because of the uncertainty of the weather for a cross-channel invasion. If they had launched but the Western Allies had been forced to delay because of the weather, then the chances of success for Operation Bagration would have been lessened. Also, by waiting two weeks, they had the advantage of Nazi troops tied down on the Western front that couldn't be brought back to stop the Eastern front offensive. Another factor was deception--which the Soviet forces had become masters at. Uncertainly over the date of launch--or any kind of sudden delay--would have worked havoc with their deceptions.
For the Soviets to wait until after the Allies invaded may sound cold-blooded, but Operation Bagration was a bigger strategic factor in defeating Hitler at that point. Its success cost the Soviets a huge number of casualties, but it indirectly saved a huge number of Allied lives and shortened the war.