Is the insurer of the person that caused my wreck required to put OEM parts back on my car?

Technically the insurance company is required to put you back in the same position you were in prior to the accident, meaning they should be doing the repair with parts that are comparable to the parts that were on your vehicle at the time of the accident.  There is no requirement that they be OEM parts.  However, you can request the body shop provide an estimate that includes OEM parts.  Whether or not the insurance company will accept that depends upon many factors, including the policy which is being used to make the repairs.