Police said the relatives were found inside the family home on the Old Cullybackey Road in the town on Monday morning. A PSNI spokesperson said there was no ongoing risk to the public.
“While details continue to emerge about this incident, it is clear there has been a tragedy,” said Ms Mulholland.
“The community in Ballymena is in shock already and my thoughts are with everyone affected by what will be an incredibly distressing time, particularly the family and friends of the deceased.
“I have spoken to police, will need time to investigate this matter and I urge anyone with information on it to give it to them immediately.”