Police confirmed the incident but said it was too early to say exactly how many people were hurt or whether it was an accident or the driver had deliberately rammed the car into the crowd.
"Unfortunately, we cannot rule out that it was done with intent," police spokesman Reiner Linger, police spokesman for the nearby city of Kassel, told Welt TV.
German news website HNA cited witnesses as saying the driver appeared to have deliberately targeted children and had driven "at full throttle" into the crowd, which had gathered for the Shrove Monday procession.
Bild said that a third of more than 30 people hurt were seriously injured. Police called off all carnival parades in the German state of Hesse, where Volkmarsen is located, as a precautionary measure.
The incident comes less than a week after a man gunned down 11 people, including himself, in one of the worst racist attacks in Germany since World War Two.
Carnival is hugely popular in parts of western Germany, especially in Rhineland cities such as Cologne and Duesseldorf, where festivities peak on "Rose Monday" with tens of thousands attending street parades featuring comical or satirical floats.
Police cars and ambulances rushed to the scene in Volkmarsen, a small town in northern Hesse, 260 miles (420 km) west of Berlin.
"We are on the ground with a big deployment. An investigation is underway," north Hesse police said on Twitter after the incident, which they said occurred at about 2:45 p.m. (1345 GMT).
Amateur pictures published online showed police officers standing next to a silver Mercedes-Benz car that appeared to have been involved.
German media said the driver deliberately broke through plastic barriers set up by police around the parade area.
The car had continued driving through the crowd for about 30 meters (33 yards) before coming to a halt, an eyewitness told broadcaster Hessenschau.The broadcaster said eyewitnesses described the man as between 20 and 30 years old and that around 1,500 people had been expected to watch the parade.