I'd try to avoid midday at the outdoor sights. I think St. Peter's opens at 7 AM (not sure whether it's different on Sunday), so it would be theoretically possible for early-risers to go there first and then hit the Colosseum and Forum before it gets too hot. The difficulty is that it's very difficult to get any Colosseum tickets, and it seems the earliest time slots are most popular. The tour operators seem to have a way to grab them.
I suspect there's no need to buy a ticket to the Capitoline Museums in advance. Unfortunately, those museums are quite large, and they're up a long staircase from street level. It takes quite a lot of time just to walk through them, without actually looking at anything. Although the collection is really good, I'd consider the Borghese Gallery instead. Every visitor to the Borghese is limited to 2 hours, so it's not an exhausting place to visit. The downside of the Borghese is that you need to buy a timed entry ticket way ahead of time to be sure of getting in. And the gallery is popular, so it will not be uncrowded. Seeing the upper floor first is a way to enjoy a less crowded visit.
I'm concerned about the plan for your parents to see the Vatican Museums on Monday morning when they need to leave Rome "around noon". Are they heading for a train or to the airport? If the airport, is their destination elsewhere in Europe or in the US/Canada? For outbound transatlantic flights their airline will probably advise them to be at Fiumicino at least 3 hours before departure.
I think the Vatican Museums open to the public at 9 AM. I say "I think" because every single day for which general-entry tickets are available seems to be sold out, so I cannot see the entry-time options. Your parents certainly would not want to risk the we-need-to-buy-a-ticket line on Monday morning.
Alas, even having a ticket (which turns out to be just a reservation for entry) doesn't guarantee immediate entry. You have to line up with other reservation holders, go through security, then line up again at a ticket window to convert your reservation into an actual ticket. (Gotta love Italy.) Once you get inside, the place is likely to be a zoo for at least most of the long route from the entry point to the Sistine Chapel--and back out again. The Museums are huge, and they were massively crowded even pre-pandemic. The situation has gotten worse in that tickets are now selling out way ahead of time.
There is a way to have a more pleasant experience at the Vatican Museums if your parents are willing to spend some extra money. A basic entry ticket costs 24.50 euros per person. For 40 euros you can buy a "Breakfast at the Museums" ticket with an entry time of 7:45 AM or 8:15 AM. In addition to the breakfast, you get into the museum 45 minutes to 75 minutes before the general public. I think that's clearly worth the extra 15.50 euros for anyone with a serious interest in seeing the Museums. (And without a serious interest I'd advise skipping them.) The Breakfast tickets are currently on sale through July 10, so your date will be coming up soon. Those tickets will sell quickly, so if you want them, you'll need to stay on top of the ticket situation and possibly get up in the middle of the night to make the purchase.
Another way to get into the museums early--even earlier than with the Breakfast ticket--is to buy a Prime Experience ticket at 68 euros per person. Those admit you to the Museums at 7:30 AM and include a guided tour. Prime Experience tickets are already on sale for the period you'll be in Rome. If they are what you want, do not delay; they could sell out at any time.
Commercial tour operators sell early-access options somewhat like the Prime Experience deal offered by the Vatican itself, but they are substantially more costly.
