An easier way of saying this would be; His parents emigrated/ came to the US from Italy.

yes ,that sentence used" The completion of the past". when use this tense: first:when you want to describe the act what happend in the past and last the moment,or the act was completion before the moment .you can use this tense.for example :By the end of last term we had learned 1000 English words. second:if the sentence have these verb (hope,think,expect,mean,intend,suppose,want and so on),the mearning of the sentence is that the hope,pland and intention wasn't come true in the past. for example:i had hoped to see more of bei jing.