Perfectly elastic supply implies a situation of infinite supply corresponding to a given price. In such a situation, if demand curve shifts to the right, implying increase in demand, there does not arise a situation of excess demand because even at the existing price, supply is infinite. Hence, price remains constant.
If there is decrease in demand, when demand curve shifts to the left, there is no possibility of fall in price. Because even the slightest fall in price would mean zero supply in a situation of perfectly elastic supply curve.