NBA players are used to receiving things that might strike anyone else as odd.

There is nothing too strange to autograph and nothing too obscure to endorse. It takes something truly peculiar to surprise them—such as the contents of a package shipped to Dallas Mavericks forward Dirk Nowitzki last week.

Inside was Mr. Nowitzki’s picture on a potato.

Mr. Nowitzki was perplexed. This mail was so deeply weird—something Mr. Nowitzki had never before seen in his 19-year career—that he shared it with his millions of followers on social media.

Mr. Nowitzki’s potato

“Whoever sent me this POTATO!!!!” he wrote on Twitter. “Much appreciated.”

The potato arrived when the Mavericks were on a road trip. It still reached him after a team employee passed along a picture that Mr. Nowitzki wasn’t expecting. “I tweeted it within 10 seconds,” he said. “You don’t get a potato every day.”

He may have been the first NBA player to be sent a potato. He was not the last.

“I guess that’s the hot thing,” Mr. Nowitzki said.

That same day, Miami Heat center Hassan Whiteside tweeted his own tater, which featured a photo of Mr. Whiteside posing for a selfie with the musical artist DJ Khaled. He didn’t know who sent it—or why.

Golden State Warriors star Kevin Durant got one, too. He cradled his potato while being treated for an injury—which happened to be its exact intent. “Wishing you a quick recovery, KD!” read a message written in block letters on its back. “Hope this stupid potato eases the pain.”

Potato Portraits of Basketball’s Biggest Stars Some wacky tubers from Potato Parcel caught the attention of NBA players and their fans Golden State Warriors ‘Splash Bros’ Stephen Curry and Klay Thompson. Potato Parcel 1 of 12 • • • • • 1 of 12 Show Caption Golden State Warriors ‘Splash Bros’ Stephen Curry and Klay Thompson. Potato Parcel

The case was about to get even more curious.

Good teams got potatoes. Bad teams got potatoes. Suddenly, for no apparent reason, the NBA was at the center of a potato outbreak.

Even relatively anonymous players like Indiana Pacers guard C.J. Miles received personalized spuds. “One of the most underrated players in the game!” his read. “Congrats on passing Troy for 3 pts made 3J!” The achievement that warranted a congratulatory potato was the player known as “3J” eclipsing a former Pacer named Troy Murphy among players with the most 3-pointers in the team’s history. Mr. Miles is now seventh on the list.

The next clue might have been missed entirely if not for careful detective work. Denver Nuggets forward Wilson Chandler seemed to be heeding the message of his potato when he tweeted its photo. “Thank you for keeping it real on Twitter, Wilson,” it said. And then Mr. Chandler quickly deleted the tweet.

The league had a mystery on its hands: Who was this crazy fan sending potatoes to NBA players?

“It wasn’t some crazy fan,” said Riad Bekhit. “It was me.”

It was also a marketing stunt. Mr. Bekhit is the owner of a Bay-area-based startup called Potato Parcel. He’s in the business of shipping potatoes. They aren’t meant to be eaten—they’re greeting cards.

Customers upload images and messages, and for about $15, Potato Parcel sticks them on a potato.

Riad Bekhit, the owner of Potato Parcel, sent the potatoes as a marketing stunt. Photo: Potato Parcel

Potato Parcel’s intellectual property isn’t patented, though, and it has competition in companies like Mail a Spud and MysteryPotato, which offer a similarly absurd service. To distinguish his potato startup from the other potato startups, Mr. Bekhit has experimented with unconventional marketing.

MysteryPotato owner Jeff Kelly understands his strategy: He often sends celebrities his own product. They “are particularly good gifts for the person who has everything,” Mr. Kelly said, “because they don’t have a potato.”

It was because of another ploy for attention that some NBA players may have already recognized Potato Parcel. Mr. Bekhit and Alex Craig, the company’s founder, appeared last year on the television show “Shark Tank” wearing potato costumes to make a simple pitch: “We mail potatoes—and that’s it.”

Their potential investors were impressed. Potato Parcel’s executives struck a deal after they received two funding offers. Neither was from Dallas Mavericks owner Mark Cuban. He eventually admitted the idea wasn’t “completely crazy,” but he was more skeptical in the beginning.

“It’s just stupid on a stick, right?” Mr. Cuban said.

“It’s actually stupid on a potato,” Mr. Craig said.

Potato Parcel was seeking fresh publicity when Mr. Bekhit came up with his NBA caper. “There was no meeting,” he said. “This was an impulse decision.” His team whipped together some potatoes and shipped them to NBA arenas and team offices. Then he waited.

He first had to hope they made it to their destinations. He wasn’t too worried: Mr. Bekhit says his company has sent more than 40,000 potatoes. Next they had to survive layers of mail security. The last step was the most uncertain. He didn’t know what NBA players would think when they ripped open packages to find potatoes inside.

To reduce the risk of failure, Potato Parcel sent hundreds, one for almost every starter in the NBA.

Each potato also had a note for its recipient. Mr. Nowitzki was honored for scoring his 30,000th point. Houston Rockets guard James Harden was complimented for his beard. Detroit Pistons forward Tobias Harris was thanked for being on Mr. Bekhit’s fantasy team.

While some were sufficiently creeped, the scheme worked as Mr. Bekhit wanted: The world’s most famous basketball players were tweeting photos of potatoes.

Everyone from National Potato Council executive John Keeling to Mr. Cuban himself agreed it was shrewd marketing. “I love it,” Mr. Cuban said in an email Sunday. “Does that make them an investment I would make? No. It doesn’t scale. But it’s a great business for them.”

There was soon another twist: The Carolina Panthers said one of their NFL players was sent a potato that looked awfully similar to Mr. Bekhit’s potatoes—except he says he isn’t the source. “That one, quality-wise, wasn’t one we’d do,” he said. “As simple as it is, we do have a process.” The team didn’t respond to requests for comment; the sender remains at large.

Mr. Bekhit tracked his own spuds in transit and was notified when they were delivered. By this weekend, every potato had arrived. It was time Mr. Bekhit turned himself in.

The Wall Street Journal was able to corroborate his claims after Mr. Bekhit provided ample evidence: He agreed to reveal some of the potatoes that hadn’t been unearthed—including the one he sent to LeBron James.

“What can I say?” Mr. Bekhit said he wrote to Mr. James. “Thank you for inspiring me to be a beast in the potato biz.”