Children who grow up around substance abuse often develop unhealthy coping mechanisms, one of which is abusing substances themselves. Children learn from their parents how to behave, how to deal with stress, appropriate relationship boundaries, and so on.
If parents often use substances to cope with stress or uncomfortable feelings, their children learn this. Not all children of addicted parents will come to abuse substances, but it is a common narrative.
There are many reasons for this, though, beyond simply learning the behavior. As these children are more likely to experience trauma, neglect and abuse, they have utmost need for coping skills to process these challenging experiences. Unfortunately, if their parents are in active addiction, it is unlikely that they develope healthy coping skills.
It is probable that these children will use Alcohol or drugs early in life. With little to no healthy coping skills, many of them will use substances out of desperation to deal with their trauma.
Unfortunately, when trauma is processed improperly in this way, it is likely to create Post Traumatic Stress Disorder (PTSD). It shouldn’t be a surprise, then, that PTSD is a common underlying cause of addiction.