Governments contract with third parties for a lot of things. From keeping the roads free of snow, to building government buildings, public bodies often rely on private business to get the job done. There is no reason that these same bodies cannot rely on third parties for parking enforcement. They would likely do a more efficient job because they can service many municipalities with the same equipment.
Some municipalities have helped their budget by contracting enforcement and maintenance of parking to third party or private companies. Tickets issued by these companies are legitimate, but it is a good idea for anyone issued a ticket that seems suspicious to them to verify the legitimacy of it with their local law enforcement.
I don't see why states can't hire outside contracts in order to collect for parking tickets. It just doesn't seem fair to do so. The government should be collecting for the parking tickets. The money is going to them. It also seems kind of wasteful since the state governments have to pay for these contractors.
It appears that the government, both state and federal, does whatever they so desire. Yes, it is quite possible for them to contract the tickets to a third-party company. In fact, some states are already doing it. Some California drivers have had to pay a company named Ace Parking for tickets.