We were flooded with calls from worried parents.

were 是 be 动词的过去式复数形式，表示过去的状态。 flooded 是动词 flood 的过去分词形式，flood 本意是"洪水泛滥"，这里用作比喻表示"大量涌入"。 with 是介词，表示"伴随、带有"的状态。 were flooded with 这个被动结构表示"被...淹没/充斥"，形象地描述接到大量电话应接不暇的状态。