UPS drivers owe the public a legal duty of care to drive safely and to follow all rules and regulations. If a UPS driver has acted in a way that another reasonable driver would not have, and this resulted in an accident, then both the driver and UPS can be held responsible.
Unlike Amazon, UPS classifies all their drivers as employees, making holding the company responsible a bit easier. When a UPS driver does something wrong, the company is liable for the resulting damages.