There is a source: This academic article also recounts the incident:

In 1893 in Laurel, Indiana, for example, "an aged and well-known farmer" called Wesley Parsons was "joking with friends" when, according to a contemporary report, "he was seized with a spell of laughing, being unable to stop": "He laughed for nearly an hour, when he began hiccoughing, and two hours later he died from exhaustion."

The citation that the article gives is to:

The Star (Guernsey), 14 December 1893, quoted in Jeremy Clay, The Burglar Caught by a Skeleton, and Other Singular Tales from the Victorian Press (London: Icon, 2013), 139.

Awesomely (as an older dude, the marvels of modern technology never cease to amaze me), you can actually go look at the original report in The Star at the British Newspaper Archive. The article does indeed exist, but disappointingly it doesn't expand much on the quote above:

LAUGHED HIMSELF TO DEATH

Wesley Parsons, an aged and well-known farmer died at Laurel, Ind., under peculiar circumstances. While joking with friends he was seized with a spell of laughing, being unable to stop. He laughed for nearly an hour, when he began hiccoughing, and two hours later he died from exhaustion.

Regrettably, that's all we have. That's literally all there is to the article, and it doesn't even have a byline/author. So whether you believe the tale or not largely depends on whether you believe the 1890s Guernsey Star's "slow news day" -type filler. The more interesting question for me is how a paper in Guernsey got their hands on a story form Indiana, without it being reported in the American press for the period at all (I looked, but couldn't find anything).

As the man who published not only The Fortean Times Book of Strange Deaths (1995) but also The Fortean Times Book of More Strange Deaths (1998), I find myself uniquely well-placed to address this question. The mode of death specified for the unfortunate Wesley Parsons is not only quite possible, it is caused by an accepted medical condition, and it has happened in other documented cases. One, which occurred in the UK in March 1975, was very highly publicised at the time, and because the joke that triggered the death was broadcast on national TV, we do know what it was, and it can still be viewed now, if you are feeling brave enough.

So read on at your peril.

The condition that causes this form of death is a heart rhythm disorder called Long QT syndrome. It is an hereditary condition that "can induce cardiac arrest when triggered by exertion or adrenaline."

The unfortunate victim of the syndrome whose death received wide coverage here in the UK was a Norfolk bricklayer called Alex Mitchell, who died while watching an episode of The Goodies, a comedy team closely associated with the Monty Python troupe, but whose show involved a lot of slapstick and was generally considered to be more "for kids." The programme in question featured

Bill Oddie as a black belt in ‘Ecky Thump’ - a little-known Lancastrian martial art which involved pelting opponents with black pudding. Tim Brooke-Taylor played a Scotsman who defended himself with a set of bagpipes.

According to his wife Nessie, Mr Mitchell, 50, was in stitches throughout the episode then “gave a tremendous belly laugh, slumped on the sofa and died”. She later sent a letter to the Goodies thanking them for making her husband’s final moments so happy.

The clip featuring the climactic fight can be seen here [NOTE: contains some racial stereotyping that's pretty offensive by modern standards], and an interview with Mr. Mitchell's granddaughter discussing Long QT syndrome is here. I trust that postings to AskHistorians carry some sort of liability insurance.

The Star was a pretty po-faced paper, actually, and not too prone to invention. It was more common for papers to take these sorts of miscellaneous news items from other newspapers, and US news was shipped across the Atlantic to be disseminated in this way.

And actually the story does appear in the US press – which certainly did have a reputation for inventive hoaxing in the 1890s. The Library of Congress's Chronicling America site offers this, from the Maysville, Kentucky, Daily Public Ledger for 27 November 1893. Because Chronicling America merely samples the vast range of US local newspapers, I'd place a fairly hefty bet on this story having appeared in dozens of other small town papers at the time. Note the more exact date and the change of location:

Near Laurel, Md., Wesley Parsons, an aged and well-known farmer, was joking with friends Sunday morning [that might be either 26 November, or, if we don't trust the US press to get its out of state news so quickly, 19 November], when he was seized with a spell of laughing, being unable to stop. He laughed for nearly an hour, when he began hiccoughing, and two hours after died from exhaustion.

So essentially exactly the same story, in the same words, barring a change of state.

We may be getting a little bit closer to the source, however, when we check the Harrisburg [PA] Telegraph for 27 November and the Philadelphia Times for the 26th, both via newspapers.com. The Telegraph has the same story, but attributed to the small town of Laurel, Delaware. Though still very short, this is the most detailed version I have found. It lists Parsons' age as 70, adds the detail that he "fell from his chair, dead," and specifies a date of death, 25 November. The Times's version is the same but adds a source: "Special telegram to the Times." I'd say that all this points to Delaware as the most likely location of the original incident. Oddly, though, while the site carries two Wilmington papers of this period - the Morning News and the Evening Journal - neither mentions the death.