Reagan administration officials are sweating out the prospect of being forced to honor an IOU -- of$14,433,214.47 -- left in the wake of the Carter administration's sincere (if naive) attempt to tackle that most sacred of all federal cows: the free parking space.

The issue is before the U.S. Court of Appeals here. If the government loses, it could be forced to shell out at least $14 million, which represents the amount Uncle Sam collected, illegally it turns out, from civil service parkers from late 1979 to March of this year.

Late in his administration, President Carter decided that the government should set an example of energy conservation for the American people. The example he chose (trying to force federal workers to take the bus or car pools to work) was to eliminate free federal parking.

Well, not all free federal parking. The White House area was exempted, as one might expect, as were free parking spots for members of Congress on Capitol Hill, plus those close-to-the-terminal free spaces for senators, representatives and Supreme Court justices at National and Dulles airports.

Anyhow, with some exceptions, federal workers who once parked free began paying one-half of the rate charged by commercial concerns near the agency. Locally those rates ran from $12 a month to $45 a month. Civil servants were supposed to begin paying full fees by the fall of 1982.

The Carter people, who were after all, mostly out-of-towners, didn't realize how big a tiger they had by the tail when they drew up the pay parking plan. A few savvy adviser s-- Scottie Campbell of the Office of Personnel Management and Stuart Eizenstat of the White House staff -- warned that pay parking was, well, a dumb idea that might even trigger a strike in government. There were more than 150,000 free spaces involved. But hotter heads prevailed, and pay parking went into effect.

Some irate federal workers took to the streets. They picketed the White House, circulated petitions and some parked on people's lawns rather than pay the hated tariff. Those who did pay, it turns out, kept good records!

To make a long story short, the case went to court here. The American Federation of Government Employees (whose staff members pay about $25 a month to park at union headquarters) argued that the pay parking was wrong, illegal, immoral, etc. It won, on the grounds that Carter had implemented the pay parking order without the advice and consent of Congress. The government appealed the lower court ruling, but stopped that form of pay parking in March of this year. Some employes now pay to park, but they do so under a new system that covers only lot management costs (some of which are high).

The court told the government to advertise the ruling to all federal workers who had paid to park during the 17-month period before the return to free parking was ordered. And it directed the government to tell them to submit claims in the form of canceled checks, parking vouchers and receipts. The amount the workers claimed: $14,433,214, and 47 cents.

All of this research and history is presented because dozens of people called this column last week saying they had heard that the court had ruled, and where do they get the money? The court has not ruled. It has declined to say when it will. But the ruling could come next month. If it is favorable to federal workers, there could be some sizable refunds -- from $250 to $500 for individuals.