Entertaining though it is to imagine frustrated Roman drivers jockeying for spots in vast and dusty expanses (“Did you park in the Titus lot?” “No, we got stuck in Domitian…”), there is no evidence for anything like a modern parking lot in the city of Rome, for several reasons.
First and foremost, wheeled vehicles (and eventually horses) were banned within cities for most of the day. The ban began with Julius Caesar, who prohibited vehicular traffic in Rome for the first ten hours of the day (i.e., until late afternoon), exempting only public construction wagons and the carriages of certain magistrates.
Claudius imposed a similar measure on provincial cities, and Hadrian extended the ban to horses. As a result of these measures, the closest equivalents to parking lots in most Roman cities were probably suburban inns and stables, where travelers could leave their vehicles and horses for a daily fee.
Even before Caesar’s ban, few urban Romans maintained horses. There were exceptions, of course – one haughty aristocrat even kept an elephant, which he rode to dinner parties – but they seem to have been relatively rare. Even the richest Romans found it socially and politically seemly to be seen walking through their cities, a crowd of clients and dependents trailing behind.
The use of litters was thus limited, but those who were sickly or uninterested in publicity sometimes used them. (In fact, if you could afford to keep eight or so slaves as bearers and another few to clear the way ahead, litter was by the far the most comfortable way to travel, and probably the conveyance of choice for elderly senators.)
So – where were all the litters parked during games at the Colosseum? There does not seem to have been a dedicated space. Games at the Colosseum, it should be remembered, were fairly infrequent, and in a city as cramped as Rome, it was unthinkable to waste space on anything so frivolous as a parking lot for occasional events.
It is possible that litters could be left, presumably for a fee, in the cavernous substructures of the Temple of Venus and Rome (where equipment for the Colosseum was kept), or squeezed into some park or square. But we guess that people who arrived in litters simply sent their bearers back home, and instructed them to return at a certain time. Since every litter in a public place was surrounded by a gang of burly litter-bearers, theft was not usually a problem.