Sakura lives with her mother and father and is an only child. She most likely lived a normal life with her parents, no different than anything one would experience in the real world. In fact, looking at how she acts, she was probably even spoiled. ^^ It's never said what her parents think of her being a shinobi, but they are probably proud and supportive of her choice.
We never meet Sakura's father and we never see Sakura's mother, although we hear her talking on a few occasions, usually asking Sakura to run errands for her. Sakura usually responds as any typical teenage girl would; with distaste and annoyance. Later on, she seems to act more reasonable towards her mother's request to run some sweets over to the Yamanaka household, which seems to suggest that she has grown up a bit more.
Aside from this information, we don't know much about Sakura's parents or her relationship with them, which is surprising since we know so much about Sasuke and Naruto's family (or lack there-of).