When two people have been living together and they decide not to live together anymore, they are separated. However, when married people separate, their marriage has not yet ended. They have to get a divorce to legally end a marriage. Common-law couples don't have to get a divorce, because there is no marriage to end.
Parents who chose to live together without getting married don't have to get a divorce, because there is no marriage to end. But they do need to decide what will happen to their children and how they will divide their property.
Ideally, your parents will make the decisions together about who you will live with and how that will work. Your opinion should be taken into account.
If they can't decide themselves, they might go to a mediator for help in reaching an agreement. Or they might have to go to court and have a judge make the decisions for them.