An accident can have more than 1 cause and both drivers had a duty to see what was there to be seen. They probably each have some culpability.
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New York is a comparative fault State, meaning that fault can be shared by the parties to an accident. In other words, it can be split 50/50, 80/20 or even 99/1 (although that would be unusual).
Sorry to say, but the general rule is that the person driving on the roadway has the right of way and someone pulling out of a parking spot must yield until they are clear. However based on what you describe the car changing lanes may share some of the blame. I would expect some sort of split. Hopefully your father has collision coverage otherwise he may have to bear the some of the cost of repairing his vehicle.
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It sounds like lions share of the fault would lie with your father as the operator of a vehicle entering the roadway from a parking spot. Vehicle and Traffic law section 1143 requires a driver entering the roadway to yield the right of way to all approaching vehicles on the road. Section 1162 prohibits a parked vehicle from moving until it can move with reasonable safety. It sounds like the insurance company is, justifiably, of the mindset that your father is at fault for violating one or both of those sections of the Vehicle and Traffic Law.
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As already stated, the general rule is that the driver on the road has right of way and another driver entering the flow of traffic has the duty to yield. In the absence of any evidence of fault, therefore, an insurance carrier can only conclude that your father was at fault. Here, however, there is very compelling evidence: the other driver's admission! Your insurance carrier couldn't hope for more straight-forward, unambiguous, probative evidence of who was responsible for the accident. Therefore, your carrier has no basis for blaming your father, denying any claims, or raising your premiums. Call your claims adjuster and ask outright how they can ignore the other driver's admission. Hope this helps.