Sorry to disappoint you, internet. The Young Pope, while awesome in many ways, is not based on a true story. But that doesn't mean that there aren't aspects of the show that are based in reality. For example, there was no real Lenny Belardo, and for that matter, there's never been an American Pope. But the show uses the Vatican's history and real conflicts to shape its story. Similarly, though The Young Pope's Cardinal Angelo Voiello isn't based on a real person specifically, his position does exist within the Vatican.
As played by Silvio Orlando, Voiello is not happy about the Pope's unorthodox methods of ruling and his "out there" antics. This should lead to a lot of conflict, as the Vatican or Cardinal Secretary of State is a real position considered to be one of the Pope's most trusted advisors. The current Vatican Secretary of State is Cardinal Pietro Parolin, who has held the position since 2013. The job is one of the most senior offices in the Roman Catholic Church, as they advise the pope directly and can even represent him. So if Voiello and Pope Pius XIII continue to butt heads on The Young Pope, that could pose a major problem.
Pope Pius' election as the first American Pope to lead the church comes as a surprise in the premiere, but really, it's Cardinal Voiello who has masterminded the election in order to put a new face forward, so he could keep pulling strings behind the scenes. The relationship between the two has been the central conflict so far, largely due to Lenny's views. In real life, however, this relationship doesn't seem to have the same dramatic nature.
Cardinal Parolin has often spoken on behalf of Pope Francis and they appear to be a united front, which is unlike the relationship displayed in The Young Pope. Parolin has also been open about his views, talking about his hopes for an improved relationship between the Holy See and China, and even telling reporters that he believes a woman could hold his position someday.
There aren't many similarities between Voiello and Parolin beyond holding the title of Vatican Secretary of State. As is the case with many TV series, The Young Pope is telling an interesting, dramatic story that is only rooted in the most general of truths. It adheres to the real hierarchy of the Catholic Church, but the characters who make up that hierarchy are works of fiction. Perhaps Carinal Parolin will share his thoughts on The Young Pope eventually, but for now, he exists totally separately from Cardinal Voiello.