There is an old race track gag about a horse who makes his first start at the age of 5. Most people assume that his belated debut must be due to infirmities or ineptitude, but when he wins by 10 lengths everybody asks the trainer, "Why didn't you run this horse before?"

"We couldn't," the trainer says. "He was so fast we couldn't catch him until now."

Bettors at Delaware Park may be moved to think of this story this afternoon. When they study the past performances for the first race, they will see something they have probably never seen before. In this otherwise unremarkable maiden claiming race, a gelding named Pete's Fool will be making the first start of his career at the ripe age of 10.

Hasn't the trainer been able to catch him?

Not exactly, owner Harry Best Jr., said. Instead, it took him 10 years to realize that the horse might be able to run fast.

Best, who works for a chemical company and owns a farm in Chester County, Pa., bought a mare named Miss Stifle some years ago, intending to breed her and use her progeny as hunters. He bred her to a thoroughly obscure stallion named Buffoonery--"Some judge around Malvern (Pa.) had him," Best recalled--and the product of this mating was Pete's Fool.

While his contemporaries were being prepared intensively to start their competitive careers at the age of 2 or 3, Pete's Fool was still loafing on the farm. "He never really did much until he was 3," Best said. "When you go to make a hunter, you let them develop and mature. When he was 4, he started hill-topping, doing light hunting."

Not until he was 5, an age when many of his contemporaries had retired from racing, did Pete's Fool get into serious activities. Best would ride him in fox hunts at Brandywine that sometimes last for several hours. He seemed to thrive on this regimen, and last year Best even entered him in a couple of jump races at hunt meetings.

But he never entertained any fanciful notions of trying Pete's Fool in a thoroughbred race. Best had a 4-year-old maiden whom he was training for that purpose, and this spring he planned to send him to Penn National for a workout. Norman L. Brown, who is Best's "official" trainer, because Best himself does not have a license, suggested, "Why not stick that timber horse in the trailer and work them head and head?"

Best agreed, and Brown sent the two horses to the track for a three-eighths of a mile workout. When they came back, Brown told Best, "You're training the wrong horse."

Pete's Fool had beaten his stablemate in the workout and earned the chance to launch his competitive racing career at the age of 10. "It's been quite a chore," Best said. He has tried to transform a horse who has been accustomed to galloping for hours and leaping over obstacles into one capable of running six furlongs in 1:13 or so.

Now, he thinks, Pete's Fool is ready, and the gelding is entered today against a group of $5,000 claimers. The Racing Form insultingly lists his age as 12, but he is still spotting from four to seven years to his rivals.

"I wouldn't be putting him in there if I thought he was going to look bad," Best said, "but I have no idea how he's going to run." But very likely, when the race is over, nobody will need to ask Best if Pete's Fool was too fast to catch for the last 10 years