If you are concerned about the pigeons having bugs, then I would take them to the vet and get them looked at. You can also call a shop that deals with birds, and talk to them about it. I really don't know one way or another, but it sounds like a scare tactic to me. Hope this helps.
Most wild birds carry mites that are not zooalogical parisites which means bird mites stay on birds. But there are many types of mites and some don't care what species they live on. All birds can get fleas. Fleas is recognizable by "flea dirt" This looks like dirt but is actually flea feces. That's about my knowledge of bird parasites. You should contact a vat specializing in birds.