Hi Maelle,
thank you very much, did see this only today, am already back home. Yes, I do participate on ornitho.
For any future visitors maybe my experiences will be helpful:
I spent some hours at the Borghese park, lots of Italian Sparrows, Hooded Crows, Feral Pigeons, also some Blackcaps, Blackbirds, Serins, Moorhens, Yellow-Legged Gulls, Wagtails, Robins, Great and Blue Tits. Rose-Ringed and Monk Parakeet are also easy to see. Dipped on Sardinian Warbler.
Park Doria Pamphilj and Parco di Caffarella were also recommended to me. Did not manage to get to the second one due to lack of time. Doria was quite nice but I was there in the late afternoon and the place was crawling with people so I did not spend much time.
Vasche di Maccarese (the place linked to by Maelle) is awesome! The homepage in English does not give too much info, and nobody there responded to my mail inquiry. I contacted an ornitho member, and he was incredibly helpful and arranged a visit there for me. The place is not open to the public so you need to make arrangements first! I went there by taxi from the airport (not too far but not easy to find - make sure you have a good map for the taxi driver). Do not know if and how to get there by public transport.
Anyway, saw more than 50 species at Vasche, including stuff like Water Rail, Spotted Crake, Little Crake, Spoonbill, Spotted Redshank, Ruff, Green Sandpiper, Sheldduck and much more! Unfortunately the weather was pretty bad, but the place itself is definitely to be recommended, and a good option for any Rome visitor!