It hit crowds watching the Krewe of Endymion parade in the Mid-City district early on Saturday evening.
One person was in custody on suspicion of driving while intoxicated, police said, and terrorism was not suspected.
The driver of the pick-up truck seemed unaware of what he had done, one eyewitness told local media.
"He was just kind of out of it,'' Kourtney McKinnis told the New Orleans Advocate.
Police Chief Michael Harrison said 21 people were taken to hospital, five in serious condition, with another seven also injured.
A police officer was among the injured, he said.
Social clubs, known as Krewes, organise different parades in New Orleans during Mardi Gras.

Accountancy firm PricewaterhouseCoopers (PwC) said security has been beefed up at the residences of Brian Cullinan and Martha Ruiz following the mistake.
Celebrity website TMZ.com said the pair were in fear of their lives.
On Wednesday it emerged they will not be employed to do the Oscars job again.
But TMZ.com reported that they will not lose their accountancy jobs.
PwC spokeswoman Carey Bodenheimer said that they had been given protection after their home addresses and photos of their families were published in the media.
This is why both PwC accountants have been banned
How did the Oscars mistake happen?
Nine epic awards fails
On Sunday La La Land was mistakenly named best picture instead of winner Moonlight.
The team behind La La Land were interrupted mid-acceptance speech before the real winner was revealed.
It has been described as the biggest mistake in 89 years of Academy Awards history.
Mr Cullinan mistakenly handed the wrong envelope to the two presenters, Warren Beatty and Faye Dunaway.
They were given the back-up envelope for best actress in a leading role - rather than the envelope which contained the name of the winner for the best film.
PwC counts the votes and organises the envelopes. It has apologised for the mix-up.
Academy President Cheryl Boone Isaacs on Wednesday said that the relationship with PwC was now under review.
Mr Cullinan tweeted a picture of best actress winner Emma Stone minutes before handing the presenters the wrong envelope, and Ms Boone Isaacs blamed "distraction" for the error.

Ospreys players Olly Cracknell, De Kock Steenkamp, Dan Baker, Dafydd Howells and Sam Parry joined collectors in Baglan Moors on Tuesday.
The players said it was to show their commitment to the "Ospreylian community".
Last week they took part in collections in Swansea and Bridgend.
Cracknell said: "It was an early start but a great work-out for us this morning.
"A huge thanks to the recycling teams for showing us the ropes, they do a great job every day and we've enjoyed getting involved and giving a hand."
A vehicle has driven into a crowd at a Mardi Gras parade in the US city of New Orleans, leaving 28 people injured, some seriously.
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The two accountants who muddled up the main award envelopes at Sunday's Oscars ceremony have been given bodyguards following reports they have received death threats on social media.
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They are more familiar with recycling the ball than rubbish but five rugby players took a break from the pitch to collect refuse in Neath Port Talbot.