There are no solutions to this contradiction and every position is a compromise.
In general, birds that are large have less to fear from predators than those that are small and birds that flock have less to fear from predators than solitary birds because there are more eyes to spot danger.
Parrots have an advantage in this respect which makes them suitable contenders for some striking apparel in the plumage line.
Also, where many related species occur together, as in parrots, it becomes particularly important that each species be promptly recognized for what it is: if there is confusion among the sexes, this will inevitably lead to missing out in the mating stakes.
Conspicuous markings and distinctive colors are therefore an advantage when it comes to propagating the species.
So, taking all factors into account, parrots do not need to fear predators but they do need to be concerned about mixing themselves up, so their particular compromise leans towards the flamboyant rather than the unobtrusive.