ONE OF the biggest mysteries in Washington is that although President Reagan's administration has put a freeze on government hiring, and Carter appointees have been summarily dismissed, and civil servants have retired, there are still no more parking places for government employees than there were before.

The Reagan people are befuddled by this.

One Reagan official has been studying the problem and trying to come up with the answer.

"I have a theory, though I can't substantiate it, that many Carter administration people took their parking places with them when they left," he told me.

"But that's illegal," I said. "A parking place is government property."

"I know that," he said. "But every time I go into a government garage there is a car in every place. The parking places all seem to be accounted for."

"Maybe the Carter people lied to you about how many parking places were available in the first place. In that way they could have taken one with them -- without your knowing about it."

"I've thought about it, and I've asked the FBI to run a check on the Carter people to find out if they have a parking place somewhere else that doesn't belong to them. But the FBI says they have no mandate to track down stolen parking places. Besides, there is nothing stamped on a parking place to indicate it is government property."

"Maybe the Carter appointees are still using their parking places in government buildings while they're job hunting," I suggested.

"We thought of that too, but unless the car has a Georgia license plate, we can't be sure if the vehicle belongs to one of us or one of them."

I tried to think of some other explanation. "I don't know if this could be the problem, but Republicans are more affluent than Democrats, and it's possible they aren't car pooling like the Democrats did. If every Republican drives to work alone, then you might be filling up three spaces for every one the Democrats used."

This got him very angry. "You would like to think that, because it fits your image of Republicans. But I know many Republicans who are willing to car pool. The only problem is they live in areas where it's very hard to find their own kind to pool with."

"Why doesn't President Reagan freeze everyone's parking place?" I suggested. "That way, no one could get someone else's spot. If someone quits, retires or is fired, you could just put a limousine in its place."

"What kind of limousine?"

"The ones that are reserved for Mr. Reagan's friends when they come to town."

"We might do that, but it won't solve the mystery of why, when you make the government smaller, the demand for government parking places is constantly increasing."

"OK, I'll tell you why. When the Democrats controlled the government, there were so many employees that most of them parked outside in the streets. As you keep cutting down the size of bureaucracy, more and more employees have an opportunity to park inside. As soon as one department hears that another department has been eliminated, they immediately take over their parking places."

"That makes sense, but why then aren't there more parking places available outside than there were before?"

"Those places are being taken up by lobbyists. Every time you have a new administration, you find a lot of lobbyists' cars around government buildings. They have to get to know the new people."

"I hadn't thought of that," he said. "Then you don't think the Carter people took their parking places with them?"

"Even if they did, I wouldn't be too tough on them. You may do the same thing with your parking place when you leave in four years."