Are y'all good with a super fast paced trip??? Is this anyone's first trip to Italy (or to Rome or Florence)? You mention the big/main tourist attractions as things you want to see, so I'm assuming you're trying to see a lot in these cities.
Staying in 6 cities in 13 days is super, super quick. It may feel a bit "if this is Tuesday, this must be Belgium." If that's what you want, go for it. But just know that you'll be barely settling into each city before you need to pack up and move on to the next one.
I prefer to limit a 2 week trip to 3, maybe 4, different places. I find it easier to stay longer in a place and do day trips. Doing that gives you fewer times you have to pack up and move all your stuff, plus the longer you stay in a certain location the more it starts feeling like "your" little corner of a foreign city -- you start to get to know the area and find "your" grocery store, etc. Maybe not a big deal to everyone, but I love to feel like I'm really living somewhere, even if only for a couple of days.
Cinque Terra is so gorgeous, but I agree -- the recent pictures of crowds make it seem less and less appealing. I'd skip it for this trip - you can always go back and visit it some time when it's the off season.
I'd suggest -->
Weds 5/15 - arrival in Rome
Thursday 5/16, Friday 5/17, and Saturday 5/18 - Rome
5/19 - travel Rome to Florence
Monday 5/20, Tuesday 5/21, and Wednesday 5/22 - Florence, day trip to Siena
5/23 - travel Florence to Parma
Friday 5/24 - Parma
5/25 - travel to Nice
Sunday 5/26 and Monday 5/27 - Nice
5/28 - fly home
That gives you 3 days in Rome, which feels like the bare minimum to really see it. You need a full day just for the Vatican. Then the Colosseum and Roman Forum will easily fill another day, then that leaves you with another day to see everything else - Trevi Fountain, Spanish Steps, Borghese Gallery, etc. If you really felt like that was too much time, you could always plan a day trip -- Pompeii is an extremely long and ambitious day trip from Rome, but it is possible. Ostia Antica is a different preserved ancient Roman city which is a lot closer.
The main sites in Florence can be seen in 2 days, then a day trip to Siena would be relatively easy.
I've never been to Parma, but it sounds like you've found some foodie-type things to do there. If 1 full day isn't enough for that, you could take a day from another city to add more time. Also, Parma is relatively close to Florence, so you should be able to schedule a couple of things for your travel day the day you arrive.
The day your flight arrives (assuming you're coming from the US), you won't be up for much. I like to use that first day to just wander the city and sort of get my bearings and get a feel for it. I don't love hop on/hop off busses, but they are useful for that sort of day where you just want to zone out and sort of get familiar with a city.
The travel days between cities should leave you with at least half a day to tour around and see things. Unless you guys can all get up and checked out and to the train station super early, assume you'll arrive around lunch time at the new city.