The contrast used in MRIs is not iodine. Iodine based contrasts are used in CT scans.
But if you had a serious reaction to a contrast then, depending on the situation, the doctor might recommend either a scan without contrast, or a different kind of scan. (CT rather than MRI, or vice versa, or maybe an ultrasound or x-ray would be sufficient.
Your serious reaction is of COURSE in your medical records, (even if your parents didn't keep a record, the hospital/imaging center and the ER and hospital have records). Your doctor would know about it, but as an educated adult, it's also YOUR responsibility to make sure she knows about it. (I've had many such scans and 'have you ever had a bad reaction to a contrast agent?' is a standard question on the pre-imaging paperwork.
EDIT: If your grandfather is still living, you can ask him. If not -- a life-threatening reaction like this is certainly worth doing some hunting for information. Are their multiple hospitals in the city you were in? If so, you can call them, explain the situation, and ask if they have records. (They are required to keep such records for many years.) Your current doctor can also ask look for the info. [You would need to sign a form allowing for access to/release of your medical records.])
EDIT: #2: Severe reactions ARE much more common with CT contrast. Given that you remember so little about the procedure you had before, sounds likely that it was a CT and not an MRI.