Hit a car parked on an uphill, which then pushed into the car behind me

The middle car is mine.
The car in front reversed into my car, and the impact pushed my car into the one behind me, causing a multi-car collision.
It seems like they might have confused the brake and the accelerator.
The front car's rear bumper is totaled, and the plastic parts of my front bumper are broken.
The rear of my car has scratches on the bumper and the area above the bumper.
Is it okay to ask for a full replacement for the front bumper and a cash settlement (unrepaired damage compensation) for the rear?
What's strange is that the other party's insurer is AXA Insurance, and their representative showed up but didn't even take a proper look before leaving.
The scratches on the rear are on the bumper and an area that isn't the bumper, but they just glanced at it and said they'd file the accident report and left.
Furthermore, the car behind me is a taxi, and since the driver was resting in the vehicle, he is requesting both property damage and personal injury claims.....
The driver is going too far with the personal injury claim.