I doubt that there are specific dates, other than late spring to early summer. It also depends on the water temperature, and I am sure that fluctuates from year to year slightly.
Carp spawn in late spring to early summer, over aquatic vegetation. They may choose a shallow, weedy bay. After rains have swelled their home river over its banks, they may move into flooded fields to deposit eggs on submerged plants. The splashing of their spawning commotion in shallow water can often be seen and heard
Carps and Minnows
Bowfishing -- which can be done with a compound or crossbow -- is legal in Pennsylvania only for carp and suckers. Fish can be taken day or night year round except in approved trout waters during the closed season.
The best fishing, though, lasts just a few weeks in the spring, Campbell said, when fish move into shallower water to spawn.
Bowfishing still a niche sport in Western Pennsylvania
"It's part of the carps' courtship," said Lorantas. "You'll hear them or see them splashing about in the shallows and swimming in circles when water gets about 63 degrees. That usually lasts until the water's close to 80."
Reconsidering the common carp
Carp spawn in shallow, weedy water in the spring. After spawning time, they may be found almost anywhere. Large river systems provide excellent carp habitat, but many lakes support resident populations of trophy carp. It's easy to locate carp in the spring, when they often leap from the water in the throes of their spring spawning urges. Later in the year, look for current breaks in rivers - carp are bound to be there.
Species Profile: Common Carp, Cyprinius carpio
This report says its June when the water is between 15-20 Celsius.