Who were the parents of our parents, where and when they were born, and how many children they had?
We have two available sources to know all of that: the parish records (made by the churches or parishes) and the records made and conserved by the Italian State (Civil Register = stato civile). They have the necessary information about births, marriages and deaths.
Parish records were used since long time ago, before the civil records. Through the Council of Trent, at the end of the 16th century, the Catholic Church established that every parish priest would maintain records of every parish baptisms, marriages and deaths of all parishioners and outlined the procedures for those sacraments.
Archives of Dioceses - How to locate Dioceses and Italian Parishes?
In some Diocesan archives there are duplicates or the parochial records the bishops used to send and still send to be filed.
Italy Strategies for Locating Births, Marriages and Deaths - Wiki FamilySearch
The French Revolution achieved in 1789 the separation of the Church and the State, and in 1792 the Civil Records were ruled, reaffirmed by the Napoleon Code, source of inspiration of all European legislations.
The Civil Register was implemented in Italy by Napoleon I, when he was crowned as the King of Italy (1805). At the time Napoleon was expulsed, Northern States in Italy gave those records back to the parishes again.
Yet, not all is that simple. The French left behind them their influence and several Italian regions, at the moment they returned the records to the parishes, demanded for some corrections. For instance, parishes must make annual reports, as indexes, and to report them, once a year, to the civil authorities.
These topics are fundamental:
Italy Civil Registration - FamilySearch Wiki
Civil Registration - General Historical Background
What are Processetti? - To marry in southern Italy, you needed the consent of your parents. If your father was alive he gave his formal consent before an official of civil registration. If your father was dead, however, you had to prove that he was dead; this proof you provided with a copy of his death record...... (by Robert Angelo)
Italian Allegati - Allegati are basically records that were required to be presented by a couple who intended to get married... (by Nuccia - Gente di Mare)