score:5
Peru is on the list of countries whose nationals do not need a visa to enter the Schengen area (relatively recent addition, after most of South America, if memory serves). Generally speaking, if you can enter Spain without a visa, you can almost always visit other Schengen countries (including Italy).
If your father is travelling with you from Spain to Italy, there shouldn't be any police check and I would not expect any questioning whatsoever.
If your father is travelling straight to Italy to join you there, presenting an invitation letter from a Spanish resident would be a bit odd. Instead, he should just explain the holiday plans you have and perhaps have some documentation regarding the accommodation you arranged (hotel booking, etc.)