That website is official but doesn't work for me during hours, Rome time, that the Colosseum is closed. I have the same issue with the Borghese's home site. However the ticketing site for the Colosseum/Forum/Palatine, Coopculture, which is linked to the Parco's info site, is available 24/7.
https://www.coopculture.it/en/tickets/index.html
Tickets are very difficult to get as they often sell out within minutes of becoming available. Even the outside tour companies are having issues getting their hands on them. There are any number of previous threads on the subject such as these:
https://community.ricksteves.com/travel-forum/italy/colosseum-tickets-95ef60a3-0053-49f4-b26d-15f749d8bbd9
https://community.ricksteves.com/travel-forum/italy/colosseum-forum-palatine-hill-tour-canceled-6-days-out-by-the-roman-guy-tours-any-options
Do note my mention, in one or both of those threads, of the NEW onsite ticket office. If using it as a backup, plan to get in line EARLY, well before 9:00, as they only sell tickets until that day's quota is gone, and I'm sure it'll be come a popular option VERY quickly if it hasn't already. I've no idea if that office sells tours of the underground or arena, possibly just general, timed-entry tickets. Honestly, if you can get your hands on those, I sure wouldn't hold out for an attempt to get arena or underground tours: at least you'd get INSIDE, and you can see down into part of the underground from regular tourist areas.