It may come as a surprise to many an exhausted mother or father--but thinking about your children could improve your memory,a study suggests.
这里的 an 和 but 要如何理解？
many a + 单数名词 = many +复数名词。前者为古体使用方法，目前用则看上去过于正式。
but 为并列连词，后接并列分句，thinking about your children为动名词短语作后一分句的主语。