Nothing will happen to the ownership of the home. The only important thing is who is on title of the property. Since it is only the parents, there is nothing to worry about should a break up occur. The non family common law individual has no legal right to the ownership of the property.
Now if you are buying a home together with a common law spouse and your parents are giving a rather large "gift" for the down payment, you may want to have your parent register that on title so that if a break-up does occur, it will need to be paid back. That way, the non family member of the common law relationship will not obtain the full benefit of the gifted funds. I have seen this many times