Morning vocalizations are a pretty normal part of having a conure--and many other parrots for that matter. My Sun conure used to be much noisier initially--but because I did not reward her screaming with my attention (yelling at them to shut up is a reward, as they perceive it as a contact call) her vocalizations have reduced greatly. Now she only really screams if she can hear us, but can't see us--or if we leave the room without taking her. You'll never be able to stop the calling, but you can certainly do things to lessen the amount.
Covering them with a blanket gives them some security and lets them know when it's "bed time". I find that a set schedule for sleeping (we put Newman to bed at 9pm and wake her up at 10am most days) gets them into a routine where they have expectations about when they'll receive attention. This, in my experience, limits the contact calls dramatically.
