With DL Plat status you have access to all lounges on international itinerary. I am not familiar with current FCO layout to say whether Piazza di Spagna is accessible, but Hangar lounge definitely will be.

On the way back, Brindisi doesn't seem to have a lounge available, in FCO you will have access to Hangar (physically at least) and Piazza di Spagna.

In both cases you will have access to whatever lounges you can enter via credit cards, PP etc. depending on what you hold.