I will make two assumptions that can be supported by canon, but which are not required by it.
1. During Occlumency lessons, Harry saw one of Snape's memories where a hook nosed man was yelling at a woman while a small dark-haired boy cried. I will assume the man was Snape's father, Tobias, the woman was Snape's mother, Eileen nee Prince, and the boy was Severus Snape himself. This may not be the case. The man could have been some other relative.
2. I will assume that Spinner's End was Snape's childhood home. I assume that this is the home that Harry saw in Snape's memory of the yelling man and Snape's memory of killing flies in the bedroom. This is not necessarily true. Perhaps Snape did grow up in a "Snape Manor" environment and the house on Spinner's End was bought as an adult on a teacher's salary. And perhaps he used a second hand Potions text for some reason other than not being able to afford a new one.
I base the idea that Tobias and Eileen had a good relationship with each other and with their son on several things:
1. JKR's names are usually significant. There is a book in the Bible called Tobias, in which Tobias ends up marrying a woman named Sarah, which means "princess." It's a sweet story and the implication is that the marriage was a happy one. Tobias Snape married a woman named Prince.
2. JKR has a tendency to misdirect by giving one hint in the wrong direction, one hint in the right direction and then remaining completely silent otherwise to let us draw our own conclusions. She did this with Snape's blood status. The hint in the wrong direction was calling Lily a Mudblood. The hint in the right direction was mentioning a "matchbox" in the dueling club with Lockhart in Harry's second year, which implies familiarity with a Muggle environment. (Remember that Arthur had to have "matches" explained to him.) It is rather shocking to go through every Snape scene and realize that there is not one case in canon, other than calling Lily a Mudblood in the Pensieve memory, of Snape taunting or belittling someone based on blood status. And this is a man that has no compunctions about taunting or belittling his students, especially the ones he doesn't like.
3. Snape was giving Harry Occlumency lessons because they knew that Voldemort was sometimes present in Harry and "seeing" what Harry saw. As part of the lessons, Snape was taking the risk that Harry, and by extension Voldemort, would gain access to his memories. The memory of his father that Harry saw would be the kind of memory Snape would want Voldemort to see, especially if he cared about his father and his father was still alive.
4. The memory of the yelling man is not very extreme. Note that the man in the memory is yelling, not hitting. It may not be good, but even in reasonably good families, parents will sometimes yell at each other and if a small child is present, he may cry. We have no direct evidence (only the indirect evidence of Snape's temper and general nastiness) that this kind of scene was a common occurrence. In fact, there are several characters in canon that were abused: Harry, Merope and Sirius Black, and in each case we get much more than a single brief fragment of a memory, taken out of context. (The screaming portrait in Grimmauld Place gives us strong evidence of the kind of verbal abuse Sirius was subject to growing up. And of course, he was actually thrown out of the house and disinherited.)
5. Finally, I believe that Slytherin House will have to be redeemed in the final book. I'm hoping we will be shown that Voldemort twisted Slytherin ideas about blood status into the deadly racism we see in many of the members of that House. And it would be fitting to do that, at least in part, by having the Head of Slytherin House turn out to be a half-blood with a Muggle father whom he loved.