'Carer' is a legal definition. You are a 'carer' if you spend 35 hours or more per week looking after someone who is 'severely disabled', with 'severely disabled' defined as:
eligible for, but not claiming, or claiming:
Attendance Allowance, Disability Living Allowance - the middle or highest care rate, Constant Attendance Allowance at or above the normal maximum rate with an Industrial Injuries Disablement Benefit, Constant Attendance Allowance at the basic (full day) rate with a War Disablement Pension, Armed Forces Independence Payment (AFIP), or Personal Independence Payment daily living component.
So, I'd say that you are a parent, first and foremost, but once you are caring in a way that is significantly beyond that required for typical children/young people/teenagers/adults of your child/young person/teenager/adult's age, then you are also a carer.
I refer to myself as DD1's carer because it reinforces the point that the luxury of choice is not ours - she needs care and that prevents me from working. I think 'Stay at Home Mother' implies that a choice has been made.
Perhaps I would have chosen to be a SAHM anyway? I'll never know, now, because I can't choose otherwise. I tried to return to nursing, but had to reject my course and placement offer because the course required that I was available for all shift patterns and there was no way to fulfill that with DD1's SN transport.