Two people have been detained, including the suspected driver of the vehicle.
German authorities said a 29-year-old German man had been arrested on suspicion of attempted murder and that the motive for his rampage was unclear. A third of those wounded were children.
Seven of the injured were in serious condition, Frankfurt police Chief Gerhard Bereswill said.
District administrator and carnival spectator Reinhard Kubat said that, upon hearing sirens, he ran towards the scene, passing crying people on the way.
“There were children lying all over the street,” he told dpa.
Pubs in Volkmarsen that would normally have been full of revellers and music fell silent after the vehicular assault.
“I was at work and I got a message on Facebook messenger saying a man drove his car into people in Volkmarsen,” said construction worker Johannes Peter, who was drinking a beer on his window and talking to his neighbour across the road.
“My first thought was ‘this is a terrorist attack,'” said Peter. “Luckily it wasn’t but what happened is s–t.”
The mayor of Volkmarsen, Hartmut Linnekugel, said he was “deeply shocked” by the incident.
German Chancellor Angela Merkel and Foreign Minister Heiko Maas both said their thoughts were with the injured.
“[I wish] them, their families, and all of those who were at the Rose Monday parade, much strength,” Maas said, referring to the Monday before Ash Wednesday.
The suspected driver came from the town itself, investigators in Frankfurt said.
The 29-year-old was not known to the authorities as an extremist, security sources told dpa. Hesse’s Interior Ministry said he was injured and not yet fit for questioning.
The newspaper Bild quoted a witness as saying that the driver had been surrounded by angry people with raised fists and that he had to be protected by police.
The man had driven at high speed into the parade, the ministry said. Local newspapers reported witnesses as saying the driver had gone around a barrier and sped into the crowd.
The second detained person had apparently filmed the incident from behind the car, security sources told dpa. It was not clear whether the man was filming the attack as an onlooker or was involved more directly in the incident.
Police in the state of Hesse appealed to those who had taken photographs and videos in Volkmarsen not to circulate them and to hold back from speculation.
Police said their investigations into the motive were proceeding in “all directions.”
Volkmarsen is a small town of around 6800 residents that lies around 32 kilometres from Kassel, in west-central Germany.
The incident marks a second tragedy for Germans around the festival season. It comes just days after shootings in another town in Hesse, Hanau, that claimed the lives of 10 victims.
Carnival floats across Germany this year referenced the Hanau shootings. In the carnival bastion of Cologne, for example, a float showing the city’s famous cathedral with hanging spires and tears proclaimed: “Our heart is with Hanau.”
Carnival is celebrated predominantly in Catholic communities across Germany, with a series of parades and parties in the days leading up to Ash Wednesday.
In Cologne, 12,000 participants and 300 horses took part in the procession, armed with nearly 300 tonnes of sweets to throw out to the crowds.
In Mainz, around 9500 participants were in the 6-kilometre-long parade, including more than 2100 musicians.
No more carnival parades were allowed on Monday in Hesse for safety reasons, with police presence strengthened across the state.
“It is simply a terrible, a horrible act, carried out on people who simply and happily wanted to celebrate carnival,” said Hesse’s Interior Minister Peter Beuth on Monday evening.
Reuters, DPA, AP