Joyce Parks was struggling to afford her Kia Soul when, she says, the dealership where she had bought it pitched her an unconventional idea: Stop making the payments.

Ms. Parks, 63 years old, says employees told her that she couldn’t trade in the Soul, but that she could buy another car. To get rid of the Soul, the dealership told her, she should have the lender repossess it, Ms. Parks said.

The...