This is Sprint. On phones older than the 5c the IMEI of the device is tied to the serial number of the original SIM.
OP's father is going to need the original SIM in the phone in order for the unlock to apply if the device is a 4s or 5.
OP. Note the following…Sprint currently requires a device eligible for unlock to have been active for 50 days on their network. They may demand that you have the old iPhone active before they unlock it, despite it having been used on Sprint before.
You do not mention what old iPhone your dad has, but as Sprint's first iPhone was the 4 and 4s it cannot be older than that and since you mention a SIM card it cannot be an iPhone 4 because the Sprint iPhone 4 has no SIM card tray.
So, either a 4s, a 5, a 5c or a 5s. In the case of the 4s and 5, the international unlock is ALL they will give you. They can't unlock those fully.
Also, regarding a restore, this is only really been necessary if you had a 4s or earlier and the year was 2011. It's really unnecessary to do that now although most CSRs push it on you.
But if you want to get that "Congratulations, your iPhone is unlocked message" (as proof the phone is unlocked) then go ahead and restore through iTunes.
Last note. There is no way to tell if Sprint has actually submitted the unlock or not without testing a different SIM in the phone. It is entirely possible that Sprint will drop the ball and your dad will arrive in Italy with a locked phone. So, do your best to make sure Sprint has done it's job before your dad leaves.