Sabin's parents died when he was really young. It's not really said when his mother died, but it may have been from complications at childbirth, or something else really soon after. His father died when he was seventeen, and it was suspected that poisoning was the cause of his death. I would go as far to say that there is a strong possibility that poisoning was the cause of both of his parents' deaths.
Apparently Figaro had been at war with the Empire, but had agreed on peace. The Empire falsely accepted the offer and ended up later poisoning Sabin's father. That's the theory, anyway. It was one of many things that led Sabin to rebel against the empire and consider forsaking his Kingdom.
I'm outta here! I'm forsaking this war-sick realm for my dignity and freedom.
Sabin was largely upset at his father's death, and brought to tears. That leads me to think that Sabin was definitely close to his dad. His mother died when he was really young, so his father was the only parent he had growing up. Sabin hated the Empire after his father died, and he hated Figaro. I'm thinking that Sabin blamed the Empire (and probably rightfully so) for his father's death, and Sabin hated Figaro because they didn't mourn for his dad. When he died, all they could talk about was succession, and that made Sabin sick. His hatred for the Empire would never go away, but his hatred for Figaro probably did. It was his father's last wish that Sabin and Edgar take care of Figaro, and I think he is the reason Sabin would eventually be able to care about Figaro, and do what was right for the Kingdom.
Sabin saw his father as a loving and caring man. Later on when Edgar asks Sabin if he thinks their father would be proud of him, Sabin says to never doubt it. I think that their father was the type of father that would be proud of his sons no matter what. They both had a great deal of respect for the man. Part of what motivated them were his last desires. They treated his last wish like a ruling.
Sabin may or may not have known his mother. She could have died before he had the chance. I'd like to think that Sabin did know her, because they toast their parents later and Sabin says "To Mom," like he knew her. When his father died, though, he said that Figaro probably wasn't even sad when his mother died, and that leads me to think that he doesn't remember her death, and probably doesn't remember her life either. It's really confusing and hard to say one way or the other. One thing can be said for sure, though, and that's that Sabin loved his mom and respected her a lot. The fact that he brought her up when his father died shows how important she was to him.
The fact that Sabin and Edgar toast their parents ten years after their father's death, shows how important they both are to both of them. It's obvious they both miss, love, and respect their parents.