Fares are utterly unpredictable (by me, at least), but they do tend to bounce around a lot. In addition to a fare alert (I use Google Flights), with five tickets needed I would be checking every day manually. There's too much money on the line not to do that. I've done such monitoring for my last two trips, and for this year's central-Europe itinerary the fare for Washington DC-Munich / London-Washington DC repeatedly bounced between $800+ and $1300+. Pay particular attention between now and Thanksgiving, including during holiday weeks. I think there's a good chance you'll see an opportunity to beat the $1400 fare by quite a lot. Obviously, I cannot guarantee it.
One thing I've observed is that when you're looking for a non-stop flight on a route that often requires a connection (as I assume Chicago-Rome does), you don't want to be shopping for flights only one or two months before the trip. There may still be some decent fares on connecting flights at that point, but often by that time all the non-stops are up in the price range you are currently seeing, or even higher.
You haven't specified your planned arrival and departure points in Italy (or your specific travel dates). You don't necessarily have to fly in and out of the same city. You might be able to do Rome/Milan or Milan/Rome for the same price as a round-trip to either of those cities. That's what the websites call a "multi-city" flight. You can also look at Venice/Rome or Rome/Venice--depending on where you plan to go in Italy, of course. From my origin, Milan and Rome are often the same price, but that is highly origin-dependent, and it may not work out that way for you.
Also be aware that Italy can be miserably hot in July. That would be a greater concern for me than the crowds. It is a serious consideration if your parents are elderly. Be sure your hotels are well air-conditioned and conveniently located for taking afternoon breaks. If you have the time to do so and your parents don't have serious mobility issues, I'd consider spending a few days up in the Dolomites in a place like Ortisei. You'll be nearly certain of having pleasant temperatures there because of the altitude, whereas the rest of the country may be miserable. It's a lovely area, and there are easy walks in high-mountain meadows as well as more challenging options.