It will depend upon states laws and the insurance coverages each driver has as to who will pay for the medical bills.
If this occurred in a no-fault state such as Florida then your driver's own personal injury protection (PIP) coverage would be used for the medical bills that were incurred due to this rear end wreck.
If instead the accident happened in a tort state, like California, then your driver would be able to place a claim with the at-fault driver's insurance bodily injury liability policy for their medical expenses. Alternatively, their own Medical Payments or Uninsured/Underinsured Motorist coverage could cover the claim.
There are other insurance coverages that could cover the medical bills depending upon the situation. Such as if the at fault person was uninsured then the injured driver in the other vehicle could use their uninsured motorist bodily injury coverage if they had it. Or if the driver had medical payment as part of their policy they may use it to cover their medical costs.
You can read about these different types of insurance coverages by going to the Insurance Coverages Definitions section of our Learning Center.