Paris to Rome trains

There is no single train, there are a few with a single change in Milan. 11 1/2 hours is the shortest timewise.

It is quite a distance, and there is no direct 'through' service.

Quickest & easiest via a single change in Turin.

TGV 9247
Dep 09:45 PARIS GARE DE LYON (France)
Arr 16:18 TORINO PORTA SUSA (Italy)

FR 9575
Dep 16:30 TORINO PORTA SUSA (Italy)
Arr 20:55 ROMA TERMINI (Italy)

Not directly, you have to change in at least one of Turin and/or Milan and/or Florence.
Details are here: https://seat61.com/international-trains/trains-from-Paris.htm#Paris-Italy

We have done it...It's really too long for a single trip. We did enjoy doing the trip with an overnight along the way to break it up. Torino is a great spot to stop since you have to change trains anyway,

There is a night option:

Paris Gare de Lyon dep 19:23 TGV 6707 TGV Duplex Direction: Mulhouse Ville
Subject to compulsory reservation , Global price , Bar , space for wheelchairs

Dijon Ville arr 20:56

Transfer time 1:01 h

Dijon Ville dep 21:57 EN 221 EuroNight Direction: Venezia Santa Lucia
Sleeper and couchette train , Global price , Couchettes , Bordrestaurant
+ 1 Day

Milano Centrale arr 06:00

Milano Centrale dep 07:30 ES 9609 EuroStar Italia Direction: Napoli Centrale
Global price , Bar , space for wheelchairs

Roma Termini arr 10:29

Thello runs night trains from Paris to Milan, leaving a bit after 7 PM and arriving 6 AM. From there you can transfer to a regular train to Rome on the same ticket.

Night trains go slowly, in part to avoid arriving too early in the morning. There are no stops after Dijon at about 10 PM.

In addition to investigating trains, look at Skyscanner for flight options. From Paris, try to fly out of Orly ORY or Charles de Gaulle CDG, as Beauvais BVA is much farther from central Paris than the other two. For Rome, either Fiumicino FCO or Ciampino CIA works fine.

https://www.skyscanner.com

Like Adam said, Thello runs a night train from Italy to Paris. When I was searching, I saw 4 cities as options to embark (I was looking for a Rome - Paris route, not the other way around), but the stations are the same: Turin, Milan, Verona and Venice. In my case, I was looking for a place to enjoy the day while saving a hotel day, so I was more into Venice, but, in yours, I think you'll be looking just for the cheapiest ticket to Rome. The cheapiest option is a InterCity from Turin, 12 euros, 7h travel. Ocasionaly, I see an Intercity from Milan, too, from 19 euros, and a Intercity from Venice, 13 euros. The Intercity are the regional trains, so they're slower, a lot slower. But I know the majority of the forum usually doesn't travel on a budget, or just prefer to pay a premium from comfort, so you could look into the fast train (Freccia): 3h from Milan and slightly more from Venice, 50 euros, or a 3h travel from Verona for 36 euros.

Considering low cost flights, it depends on the day you'll be flying. The cheapiest option I saw is on a thursday, 33 euros. Add the cost to go to Beauvais (16 euros) and the bus to go from Ciampino to the Termini (5 euros), you'll have 54 euros, slightly more expensive than the cheapier train option (47 euros, Paris to Turin and the InterCity Turin to Rome), but a lot faster. However, you should add the hotel fare, since you'll need to spend overnight somewhere. And, like I said, 33 euros is the cheapiest air ticket. If you fly on a weekend, you can expect fares on 48 euros. Oh, and there's the extra cost to dispatch bags, if you have a lot of those. With a regular flight (Alitalia), it's 50 euros from CDG to Fiumicino. Plus the cost to go to CDG (11,50 on Roissybus, or 10 on the regular RER train) and from FCO to Termini (14 euros, Leonardo Express), you have a total of 75,5 euros, but without an extra cost of dispatching bags (which can cost over 20 euros, depending on your total).

I would go on train, but I always prioritize budget over comfort (if I spend less money on transport, I'll have more money to spend on attractions) on everything, and I know this is a different approach than the usual. So, it's up to you and your preferences.

To further clarify, you need not connect in Dijon when taking an overnight train. Christi probably just didn't see the 19:14 departure of the Thello night train from Paris to Milan. However, there are a few days coming up when the night train doesn't run. It also doesn't have a great service reputation (e.g., cars switched and those passengers left without bedding).