The man, who is also in his fifties, has been arrested on suspicion of murder with an alternative charge of manslaughter.
According to reports the man drove the car at high speed up a path heading into the courtyard of a residential building close to Norra Infartsgatan.
Police and ambulance services were called to the scene at around 10.20am on Friday and police soon suspected that the incident was not a case of a regular traffic accident.
Ambulance staff were unable to revive the woman and she was declared dead at the scene.
The male driver of the vehicle was found near the parked vehicle close to the scene and he was detained and formerly arrested on Friday afternoon on suspicion of murder with an alternative charge of manslaughter.
“We think it is a relationship issue. We currently have no information over the lead up to the incident but we hope to gain clarity on that issue during the interrogation,” said Calle Persson at Skåne police to the local Helsingborgs Dagblad daily.
Persson said that the man had no previous criminal record and was not known to the police.
“The only thing of note is a couple of traffic offences,” he said.